Fakultät für Mathematik und Wirtschaftswissenschaften
Consumption-Investment Problems
of a Relaxed Investor
with Partial and Insider
Information
Dissertation
zur Erlangung des Doktorgrades Dr. rer. nat.
der Fakultät für Mathematik und Wirtschaftswissenschaften
der Universität Ulm
vorgelegt von
Julia Gentner
aus Aalen
2014
Amtierender Dekan:
Prof. Dr. Dieter Rautenbach
Erstgutachter:
Prof. Dr. Ulrich Rieder
Zweitgutachter:
Prof. Dr. Robert Stelzer
Abgabe der Doktorarbeit: 17. Dezember 2014
Tag der Promotion:
13. Februar 2015
To my parents
Abstract
In a financial market consisting of a risk-free asset and several risky assets, an investor with
logarithmic or power utility functions aims to maximize the expected utility of terminal wealth
and intermediate consumption. The price dynamics of the risky assets follow a geometric Brownian motion where the investor cannot observe the drift. However, the investor has additional
information about the stock prices (insider information). Therefore, the consumption-investment
strategy has to be adapted with respect to the filtration generated by the observations of the
stock prices and the insider information. In this thesis, we model the unknown drift using a
Bayesian approach and the insider information using a random variable (drift, terminal stock
prices or terminal value of the Brownian motion) whose noisy value the insider investor knows
at time zero. Since continuously trading is not possible in reality, we discretize the model by
restricting the consumption-investment decisions of the investor to an equidistant time grid with
mesh size h > 0. Using a filtering recursion we are able to apply the theory of Markov Decision
Processes in order to solve the discrete-time consumption-investment problem with partial and
insider information. To compare the three different types of insider information we define the
value of the insider information as a certainty equivalent. A numerical example lets us suppose
that an investor prefers information about the stock prices rather than information about the
drift or the Brownian motion. Since deriving optimal closed form solutions is difficult, we investigate also the continuous-time consumption-investment problem and present here an optimal
strategy in explicit form. From this solution we construct for small h a good strategy for the
discrete-time investor with logarithmic utility functions. Moreover, we study several convergence
results for h → 0. We show that in general the expected utility of the discrete-time investor does
not converge to the expected utility of the continuous-time investor.
Kurzzusammenfassung
Ein Investor mit Log- oder Power-Nutzenfunktionen konsumiert einen Teil seines Vermögens und
investiert den Rest in einen Finanzmarkt bestehend aus einer risikolosen Anlagemöglichkeit und
mehreren risikobehafteten Anlagemöglichkeiten (Aktien). Ziel ist, den Nutzen aus Endvermögen
und zwischenzeitlichem Konsum zu maximieren. Die Preisprozesse der Aktien folgen einer geometrisch Brown’schen Bewegung, wobei dem Investor die Drift nicht bekannt ist. Der Investor
hat jedoch zusätzliche Informationen über die Aktienpreise (Insiderinformationen oder private
Informationen). Eine Konsum-Investitionsstrategie muss folglich adaptiert sein bezüglich der Filtration, die von den Beobachtungen der Aktienpreise und den Insiderinformationen erzeugt wird.
Wir wählen einen Bayes’schen Ansatz für die unbekannte Drift. Die Insiderinformationen werden als eine Zufallsvariable (Drift, terminale Aktienpreise oder terminaler Wert der Brown’schen
Bewegung) modelliert, deren gestörten Wert der Investor zu Beginn seiner Handelszeit kennt. In
der Praxis ist zeitstetiges Handeln nicht möglich. Wir nehmen daher an, dass der Investor nur
zu diskreten Zeitpunkten mit Abstand h > 0 handeln kann (h-Investor). Mit Hilfe einer Filterrekursion können wir schließlich das zeitdiskrete Konsum-Investitionsproblem mit partiellen und
privaten Informationen mit der Theorie der Markov’schen Entscheidungsprozesse lösen. Um die
drei verschiedenen Insiderinformationen zu vergleichen, definieren wir den Wert der Insiderinformationen als ein Sicherheitsäquivalent. Ein numerisches Beispiel legt die Vermutung nahe, dass
ein Investor Informationen über die Aktienpreise gegenüber Informationen über die Brown’sche
Bewegung oder die Drift bevorzugt. Da es schwierig ist, eine optimale Strategie in geschlossener
Form anzugeben, betrachten wir auch das zeitstetige Konsum-Investitionsproblem und leiten
dafür eine optimale Strategie in expliziter Form her. Ausgehend von dieser, konstruieren wir
für kleine h eine gute Strategie für den h-Investor mit logarithmischen Nutzenfunktionen. Des
Weiteren wird der Grenzübergang h → 0 untersucht: Im Allgemeinen konvergiert der erwartete
Nutzen des zeitdiskreten Konsum-Investitionsproblems nicht gegen den des zeitstetigen.
v
Contents
1 Introduction
1.1 Motivation and Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1
3
2 The Model
7
3 Financial Market
11
4 Discrete-Time Consumption-Investment Problems
4.1 Discretized Financial Market . . . . . . . . . . . . . . . . .
4.2 Formulation of the Optimization Problem . . . . . . . . . .
4.3 Enlargement and Filtering . . . . . . . . . . . . . . . . . . .
4.3.1 Enlargement . . . . . . . . . . . . . . . . . . . . . .
4.3.2 Filtering . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.3 Summary of the Enlargement and Filtering Results .
4.4 Reformulation as a Markov Decision Process . . . . . . . .
4.5 Solutions for Logarithmic and Power Utility Functions . . .
4.5.1 Logarithmic Utility Functions . . . . . . . . . . . . .
4.5.2 Power Utility Functions . . . . . . . . . . . . . . . .
4.6 Properties of the Portfolio Value and the Optimal Strategy
4.6.1 The Value of Information . . . . . . . . . . . . . . .
4.6.2 Monotonicity Results for d = 1 . . . . . . . . . . . .
4.6.3 Summarizing Remarks . . . . . . . . . . . . . . . . .
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5 Continuous-Time Consumption-Investment Problems
5.1 Formulation of the Optimization Problem . . . . . . . . . .
5.2 Enlargement and Filtering . . . . . . . . . . . . . . . . . . .
5.2.1 Enlargement . . . . . . . . . . . . . . . . . . . . . .
5.2.2 Filtering . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.3 Summary of the Enlargement and Filtering Results .
5.3 Properties of the Information Market . . . . . . . . . . . . .
5.4 Hamilton-Jacobi-Bellman Equation . . . . . . . . . . . . . .
5.5 Solutions for Logarithmic and Power Utility Functions . . .
5.5.1 Logarithmic Utility Functions . . . . . . . . . . . . .
5.5.2 Power Utility Functions . . . . . . . . . . . . . . . .
5.6 Properties of the Continuous-Time Problems . . . . . . . .
5.6.1 The Value of Information . . . . . . . . . . . . . . .
5.6.2 Summarizing Remarks . . . . . . . . . . . . . . . . .
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89
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6 Comparison of the Continuous-Time and the Discrete-Time Problems
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6.1 Continuous-Time Consumption-Investment Problems with Constraints . . . . . . . 140
6.1.1 Dual Approach for Constrained Consumption-Investment Problems . . . . . 141
vii
Contents
6.2
viii
Convergence Results . . . . . . . . .
6.2.1 Logarithmic Utility Functions
6.2.2 Power Utility Functions . . .
6.2.3 Conclusion . . . . . . . . . .
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143
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159
A Appendix
A.1 Results for Chapter 5 . . . . . . . . . . . . . .
A.2 Results for Chapter 6 . . . . . . . . . . . . . .
A.3 Auxiliary Results from (Stochastic) Analysis . .
A.4 Some Properties of Matrices used in Chapter 5
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161
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Bibliography
175
Zusammenfassung
179
Acknowledgements
183
Erklärung
185
1 Introduction
1.1 Motivation and Problem Formulation
Consumption-investment problems are a central topic in financial mathematics. In a classical
consumption-investment problem an investor with a given initial wealth and a fixed investment
horizon consumes a certain amount of his wealth and invests the remaining wealth into a financial
market. The latter usually consists of a risk-free asset and several risky assets. The main focus
of the investor lies in finding an "optimal" consumption-investment strategy.
Historically, Harry M. Markowitz introduced for the first time the portfolio problem in a mathematical formulation in the fifties of the last century (see Markowitz [1952]). In a one-period
model Markowitz used the mean-variance criterion in order to select the "optimal" investment
strategy. The fundamental idea of Markowitz was to look for a balance between the risk of a
portfolio (measured by its variance) and its expected return. In the late sixties and early seventies, the next major step in the theory of portfolio optimization was taken by Robert C. Merton
in his seminal works Merton [1969, 1971]. In contrast to the model of Markowitz with only one
trading time point, Merton allowed the investor to trade continuously in the framework of a
Black-Scholes market. Still, very popular and widely used, Merton introduced the maximization
of the expected utility of intermediate consumption and terminal wealth to judge the performance
of a consumption-investment strategy. For logarithmic and power utility functions, Merton found
that it is optimal to invest a constant fraction and to consume at a rate proportional to the
wealth.
In the field of portfolio optimization the classical Black-Scholes market model still enjoys wide
popularity. The prices of the risky assets are modeled by geometric Brownian motions with
constant drift and volatility. However, it is a strong assumption to suppose that the investor has
complete information - in the sense that the market parameters are known. More realistically,
the volatility and the drift have to be estimated out of the publicly available market data. The
investor is then said to have partial information. Whereas the volatility can easily be estimated
by the quadratic variation of the logarithmic asset prices, it is hard to estimate the drift with
an adequate precision. As the drift is not known with certainty, the parameter is often modeled
as an unobservable random variable with a given distribution - the Bayesian approach. Roughly
speaking, at the beginning of the trading time the investor draws a value for the drift and then
uses the available information to update this value. There is a considerable literature studying
drift uncertainty combined with the Bayesian approach as for instance in Lakner [1995] and
in Karatzas & Zhao [2001]. More general models with an unobservable time-dependent drift
are studied in Björk et al. [2010] and in Bäuerle & Rieder [2005]. In the latter one, the drift is
supposed to follow a continuous-time Markov chain where the current state is not observable. The
aforementioned references have in common that the portfolio problem with partial information
can be reduced to a portfolio problem with complete information using the theory of stochastic
filtering.
The investor may look for external sources of information, especially, when the drift has to be
estimated. Not publicly available information may provide a competitive advantage over other
1
1 Introduction
market players. Such privileged information is labeled as insider information. One of the most
famous examples of insider trading in Germany goes back to the year 1993. The former head of
IG Metall and Daimler-Benz supervisory board member F. Steinkühler speculated in Mercedes
shares just days before the public announcement of a share swap drove share prices significantly
higher.1 Shortly afterward, the insider trading law was adopted. Nevertheless, insider trading is a
complicated phenomenon and therefore hard to spot. Mathematical models for insider trading are
often based on the theory of enlargement of filtrations. In these models, there is an underlying
probability space equipped with a filtration F representing information available to all market
participants. To model the inclusion of additional information, the filtration G is considered
which includes the public filtration F. However, G also reflects information accessible to an
insider but not open to the public. The technique of adding the additional information at time
zero is known as initial enlargement of filtrations. This theory was presented for the first time by
Itô and considerably developed further by Jacod, Jeulin, Mansuy and Yor (see e.g. Jacod [1985],
Mansuy & Yor [2006], Yor [1992]). In the context of consumption-investment problems, the
work of Pikovsky & Karatzas [1996] is one of the first using the theory of initial enlargement
of filtrations to investigate the influence of additional future information on investment and
consumption decisions. In the paper of Danilova et al. [2010] partial and insider information
are then combined in the framework of a portfolio problem maximizing the expected utility of
terminal wealth. Using techniques of initial enlargement of filtrations and stochastic filtering
Danilova et al. [2010] convert the optimization problem with partial and insider information to
a standard portfolio optimization problem. Duality methods are applied to derive an optimal
investment strategy. A more recent work on this topic is Hansen [2013] where in contrast to
Danilova et al. [2010] the dynamic programming approach is used to solve the consumptioninvestment problem of an insider investor.
In the classical Merton problem, the optimal invested fractions of the wealth are slightly deceptive.
Due to the continuous changes in the asset prices, the investor has to trade continuously in order
to keep the fractions constant over time, which is not possible in reality. There are always some
time lags where the investor does not observe the developments of the financial market and where
he does not readjust his holdings in the assets - whether because of the presence of transaction
costs or only for more private reasons as taking some time off to relax. Supposing as in Bäuerle
et al. [2012] and in Rogers [2001] that the investor trades and observes the financial market only at
a regular basis, which are multiples of an h > 0, leads to a discrete-time consumption-investment
problem. The investor is called an h-investor or a relaxed investor. In a time interval of the length
h the asset prices may fall or rise so extremely that bankruptcy may occur triggered by short
selling or borrowing. Consequently, it is not optimal for an h-investor to short sell the assets.
In general it is difficult to derive a solution of the discrete-time optimization problem explicitly.
Thus, it is convenient to consider the continuous-time problem as it often allows to determine
an optimal strategy in closed form. By virtue of the stochastic drift caused by the presence of
partial and insider information, the optimal continuous-time strategy may result in extreme long
and short positions. Hence, an optimal continuous-time strategy is not robust with respect to
discretization - the h-investor may go bankrupt by implementing the direct discretization of an
optimal continuous-time strategy.
In this thesis, we combine the aforementioned fields of research which received attention in recent
years: consumption-investment problems with partial and insider information, and an approximation of the multidimensional Black-Scholes market with discrete-time consumption-investment
problems. To the best of our knowledge this work is the first incorporating insider information into a discretized Black-Scholes market. Therefore, the main part of this thesis deals with
consumption-investment problems of a relaxed investor with partial and insider information.
1 cf.
2
DER SPIEGEL 21/1993, Hamburg
1.2 Outline
In this combined framework many questions arise:
• Is it possible to determine an optimal consumption-investment strategy of an investor with
insider information explicitly?
• Is the optimization problem more difficult by including the consumption?
• Does an optimal consumption-investment strategy depend on the insider information?
• How valuable is the insider information for the investor?
• Does the relaxed investor have a loss of information?
• Does the value of the discrete-time consumption-investment problem converge to the value
of the continuous-time consumption-investment problem as the time lag h tends to zero?
This thesis is devoted to these questions. The novelty of this work is to characterize optimal
consumption-investment strategies for an h-investor with logarithmic and power utility functions
under partial and insider information. In contrast to the existing literature (e.g. Bäuerle et al.
[2012]) the partial information is combined with the insider information. Including the insider
information into a probability distribution allows to apply the theory of Partially Observable
Markov Decision Processes (see Bäuerle & Rieder [2011], Hinderer [1970] and Rieder [1975]).
Firstly, a numerical solution of the h-investor’s consumption-investment problem is investigated.
Secondly, we derive analytically a (good) closed form solution. For this purpose, we resort to a
discretization of a continuous-time solution. We determine an optimal closed form solution for the
continuous-time consumption-investment problem (as done similarly in Hansen [2013]). Finally,
using the continuous-time framework and introducing short selling and borrowing constraints, we
are able to construct a robust and good strategy for the h-investor.
1.2 Outline
A detailed overview of the structure of this work is given below.
Chapter 2 and Chapter 3 introduce the model with the underlying continuous-time financial
market. We consider a standard multidimensional Black-Scholes market consisting of several
risky assets and one risk-free asset. More precisely, the risky assets are modeled by geometric
Brownian motions and the risk-free asset is a money market account with a constant interest rate.
We refer to the risky assets also as stocks and to the risk-free asset as a bond. The volatility is
known to the investor and in the considered framework there is no loss of generality supposing the
volatility matrix to be symmetric and non-singular. However, the investor is not able to observe
the drift. Concerning the unknown drift we follow a Bayesian approach with a multivariate normal
initial distribution as presented among others in Bäuerle et al. [2012] or for a one-dimensional
framework in Monoyios [2009] and in Rogers [2001]. At the beginning of the trading time, an
insider investor knows the value of a random variable representing additional noisy information
about the stock prices - the insider information. The random variable is defined as a convex
combination consisting of an independent noise and the exact information which can either be
the drift, the terminal stock prices or the terminal value of the Brownian motion driving the
stock prices. We proceed differently compared to Hansen [2013] where the additional knowledge
of the drift or the terminal stock prices is disturbed by an additive noise. Considering convex
combinations benefits from the fact that the insider model includes an investor without additional
information if the scaling factor of the convex combination is set to zero. In a one-asset model
such convex combinations with information about the terminal stock price or the terminal value
of the Brownian motion are studied in Danilova et al. [2010]. In the next chapter we consider
this model for an h-investor and discretize the financial market.
3
1 Introduction
In Chapter 4 the discrete-time consumption-investment problem is investigated. We first describe the discrete-time financial market which we obtain by a direct discretization out of the
continuous-time Black-Scholes market as done in Bäuerle et al. [2012] and in Rogers [2001]. After
defining the set of admissible consumption-investment strategies, we formulate in Section 4.2 the
discrete-time consumption-investment problem in a mathematical way. In Section 4.3 we first
incorporate the insider information into a probability distribution (enlargement step) which then
allows to use a filtering recursion (see Bäuerle & Rieder [2011]) in order to include also the partial
information into a probability distribution (filtering step). The latter is defined by means of a
conditional distribution of the unknown drift given the available information (see Theorem 4.3.13)
and is often called posterior distribution of the drift. Using the enlargement and filtering step
we are able to apply the theory of Markov Decision Processes in order to solve this non-standard
consumption-investment problem. In Section 4.4 we define a Markov Decision Process with an enlarged state space consisting of the wealth, the insider information and the observation. The last
mentioned is a sufficient statistic for the posterior distribution and depends on the stock prices.
For a general utility function we characterize in Theorem 4.4.6 a solution of the consumptioninvestment problem with the help of the Bellman equation. The main results are formulated in
Theorem 4.5.1 and Theorem 4.5.3 where we present optimal strategies and value functions for
an investor with logarithmic and power utility functions. However, we cannot give an optimal
strategy in closed form. For logarithmic utility functions we obtain in the terminal wealth (without consumption) and the consumption-investment problem the same optimal investment strategy
which depends on the insider information and the observation. Hence, the consumption is without
effect on the investment decisions. Furthermore, the optimal consumed fraction of the wealth is
only a deterministic time-dependent function. The consumption-investment problem with power
utility functions turns out to be more complicated. Here the optimal investment strategy as well
as the optimal consumption depend on the insider information and the observation. Moreover, to
compare the aforementioned three types of insider information (stock prices, Brownian motion,
drift) we define the value of the insider information as a certainty equivalent which is also known
as indifference price. We conclude this chapter with several properties of the optimal strategies,
the value functions and the certainty equivalent. Due to its structure, it is difficult to derive properties of the indifference price analytically. Nevertheless, for some parameterizations a numerical
example lets us suppose that additional information about the stock prices is more valuable than
information about the Brownian motion or the drift. In the framework with only one stock we
investigate the monotonicity of the optimal strategies and the value functions with respect to
the insider information and the observation. Surprisingly, depending on the market parameters
the optimal investment strategy is in general not increasing in the observation when the insider
information is on the Brownian motion or on the stock price.
Chapter 5 describes the continuous-time consumption-investment problem. After the mathematical formulation of the optimization problem we use an enlargement and filtering step (Section
5.2) to project the asset prices onto the information filtration of an insider investor. As pioneered by Pikovsky & Karatzas [1996], the initial enlargement of filtrations technique is used
to incorporate the insider information (enlargement step). In our setting with specific insider
information we are able to obtain explicitly the representation of the so-called information drift.
Afterward, in the filtering step the uncertainty about the drift is incorporated where we use a
variant of the Kalman-Bucy filter to estimate the unknown drift on the basis of the available
information (see Bain & Crisan [2009] and Liptser & Shiryaev [2001a,b]). We are then able to
describe the continuous-time investor’s state process analogously to the discrete-time problem by the triplet consisting of the wealth, the insider information and the (current) observation (of
the stock prices). Therefore, the benefit of Section 5.2 is that the aforementioned state process
contains all the needed information and the dynamic programming approach can be used to solve
the consumption-investment problem. In Hansen [2013] a Hamilton-Jacobi-Bellman equation is
also derived, however, starting by an alternative state process (consisting of the wealth and the
4
1.2 Outline
estimator of the drift). The main results of this chapter are given in Theorem 5.5.2 and Theorem
5.5.6 where we present optimal strategies and value functions for an investor with logarithmic
and power utility functions in explicit form. For logarithmic utility functions the certainty equivalence principle holds, i.e. the optimal investment strategy is obtained by replacing in the Merton
proportion the unknown drift by an estimator resulting from the enlargement and filtering step.
Moreover, the optimal consumed fraction is a time-dependent deterministic function and does not
influence the investment decisions. For power utility functions the certainty equivalence principle
does not hold and the optimal consumption-investment strategy becomes more complex. However, if the investor is not allowed to consume, the optimal strategy simplifies considerably. We
conclude this chapter by deriving in Section 5.6.1 the value of the insider information - defined
as a certainty equivalent (indifference price). For some parameterizations an investor with logarithmic utility functions prefers information about the stock prices rather than information about
the Brownian motion or the drift (Theorem 5.6.5). For power utility functions, the certainty
equivalent is more complicated to derive unless the investor is not allowed to consume (Corollary
5.6.8). Properties of indifference prices are extensively studied by Amendinger et al. [1998, 2003],
Hansen [2013] and Liu et al. [2010]. The paper of Hansen [2013] provides a numerical example to
obtain a better economic understanding of the influence of partial and insider information on an
investor’s consumption and investment decisions.
In Chapter 6 we compare the continuous-time and the discrete-time consumption-investment
problems. Surprisingly, the discrete-time insider investor does not lose (relevant) information compared to the continuous-time insider investor - the same estimator is used for the drift. In Section
6.1 we consider the continuous-time optimization problem under short selling and borrowing constraints. There is a considerable literature studying terminal wealth and consumption-investment
problems under convex constraints. The main reference is Cvitanić & Karatzas [1992], where a
fictitious financial market without constraints is introduced in order to solve the consumptioninvestment problem under constraints using the dual approach. Also the paper of Sass [2007]
essentially follows the notion and ideas of Cvitanić & Karatzas [1992] in order to introduce convex constraints into a terminal wealth problem (without consumption) with partial information
and a time-dependent drift. For logarithmic utility functions we are able to present an optimal
strategy as the solution of a pointwise and quadratic optimization problem using results of Cvitanić & Karatzas [1992] and duality properties of convex optimization (Theorem 6.2.1). With the
constrained problem we are in the position to derive in Section 6.2 several convergence results for
the time lag h tending to zero. In general, the expected logarithmic utility of the discrete-time
investor does not converge to the expected logarithmic utility of the continuous-time investor.
The main reason is that short selling is not optimal for the h-investor. Moreover, we show that
the optimal discrete-time strategy can be approximated by the continuous-time plug-in strategy
under short selling constraints. We summarize the convergence results for logarithmic utility functions in Theorem 6.2.3 and Theorem 6.2.5. These results (for logarithmic utility functions) were
shown by Bäuerle et al. [2012] for a terminal wealth problem (without consumption) with only
partial information (without insider information). Until now we are not able to prove the same
results for power utility functions. Nevertheless, we show in Theorem 6.2.9 (together with Corollary 6.2.10) that under some assumptions the value of the discrete-time consumption-investment
problem converges to the continuous-time value when short selling is excluded. In the last section
we use convergence results of stochastic integrals, see for instance Jacod & Protter [2012] and
Prigent [2003].
Section 6.2.3 concludes this work with a brief summary of our main results.
5
2 The Model
Throughout this thesis we are given a complete probability space (Ω, F, P) equipped with
T a filtration F ≔ (Ft )t∈[0,T ] satisfying the usual conditions, i.e. F is right-continuous (Ft = ε>0 Ft+ε )
and F0 contains all P-null sets of F. We suppose the probability space to be rich enough such
that all processes and random variables considered are defined on it. Moreover, we do not assume F0 to be trivial in order to allow the investor to have partial information. We call the
filtration F the background filtration. The process W̄ ≔ (W̄t )t∈[0,T ] with W̄t = (W̄t1 , . . . , W̄tm )⊤ is
an m-dimensional standard Brownian motion defined on the underlying filtered probability space
(Ω, F, F, P) with σ(W̄u : 0 ≤ u ≤ t) ⊂ Ft .
Financial Market
We consider a representative investor with a given initial wealth x0 > 0 and an investment horizon
T > 0. This investor decides at each trading time point in the time interval [0, T ] how much of
his wealth he consumes. The remaining wealth is invested into the financial market consisting of
d risky assets and one risk-free asset. We refer to the risk-free asset as a bond B and to the d
risky assets as stocks S k , k = 1, . . . , d, with d ≤ m. The price processes are given by:
Bond price:
Bt = ert
with a deterministic interest rate r > 0;
Stock prices:
dStk
=
Stk
µk dt +
m
P
j=1
σ̄kj dW̄tj
!
,
k = 1, . . . , d,
with initial prices S0k = 1. The volatility matrix σ̄ = (σ̄kj ) ∈ Rd×m is constant and we assume
σ̄σ̄ ⊤ to be positive definite. The constant drift µ = (µ1 , . . . , µd )⊤ ∈ Rd is not known to the
investor. Thus, the investor has only partial information.
All stocks are supposed to be infinitely divisible, i.e. the investor can buy and sell fractions of
stocks and bond. The investor’s aim is to maximize the expected utility of terminal wealth and
intermediate consumption. The intermediate consumption is evaluated by a utility function Uc
and the terminal wealth by a utility function Up .
Definition 2.0.1. A function U : dom(U ) → R with dom(U ) ⊆ R≥0 is called a utility function
if U is strictly increasing, strictly concave and continuously differentiable. Moreover, we demand
a utility function to satisfy the Inada-conditions:
lim U ′ (x) = ∞,
xց0
lim U ′ (x) = 0.
x→∞
For x < dom(U ) we make the convention U (x) := −∞.
In this thesis we consider two widely used examples of utility functions:
(i) the logarithmic utility function with dom(U ) = (0, ∞) :
U : dom(U ) → R,
x 7→ log(x);
7
2 The Model
(ii) the power utility function with dom(U ) = [0, ∞) :
U : dom(U ) → [0, ∞),
x 7→ xγ
with
0 < γ < 1.
The results for the power utility function can be easily transferred to U (x) =
with the following property:
xγ
γ
for γ , 0
xγ − 1
= log(x).
γ→0
γ
lim
(2.1)
For a utility function U the Arrow-Pratt measure of relative risk aversion
RRA(x) := −
xU ′′ (x)
U ′ (x)
provides an indication for the risk preferences of the investor. The logarithmic and the power
utility function belong to the class of utility functions exhibiting constant relative risk aversion
(CRRA), i.e. the fraction of wealth an investor is willing to put at "risk" does not depend on the
level of wealth. For the logarithmic utility function we have RRA(x) = 1 and for the power utility
function RRA(x) = 1 − γ. Hence, for the power utility function RRA is decreasing in γ. We refer
to γ as the risk aversion coefficient.
Partial Information
For modeling the investor’s ignorance of the drift we follow a Bayesian approach. This means that
we consider the unknown parameter µ as an F0 -measurable random variable with a given
initial distribution. We choose the initial distribution Q̄0 as a multivariate normal distribution
Q̄0 ≔ N (µ0 , Σ̄0 ) with µ0 ∈ Rd and a positive definite covariance matrix Σ̄0 ∈ Rd×d . Moreover, we
assume µ to be independent of the Brownian motion W̄ .
All investors operating in our market are faced with partial information. Thus, we refer to an
investor who cannot observe the drift as a regular investor. The unknown drift is responsible
for the public uncertainty.
Insider Information
In addition to the given partial information setting, our representative investor has some extra
information about the terminal stock prices. We call the extra information insider information
and the associated investor an insider. Contrary to an insider, a regular investor does not have
any additional information. In order to model the insider information, we assume that the investor
knows at time t = 0 the value of an F-measurable random variable I¯T . The latter represents noisy
knowledge about an FT -measurable random variable. We consider here three types of insider
information:
• noisy information about the terminal stock prices - called stock price information;
• noisy information about the terminal value of the Brownian motion driving the stock prices
- called Brownian information;
• noisy information about the (unknown) drift - called drift information.
The noise is given by an F-measurable and multivariate normally distributed random variable ε̄
with
d
ε̄ = (ε̄1 , . . . , ε̄d )⊤ = N (0d×1 , Σ̄ε ).
8
The d × d-matrix Σ̄ε is positive definite. We suppose ε̄ to be independent of FT and therefore
independent of µ and W̄ . The components of ε̄ are not assumed to be independent. Here 0l×n
with l, n ∈ N0 denotes an l × n-matrix with all entries equal to 0. If the dimension is clear from
the context, then we simply write 0.
We define I¯T for the three types of insider information:
(I) Stock price information I¯TS = (I¯TS1 , . . . , I¯TSd )⊤ ∈ Rd , where
I¯TS ≔ aȲT + (1 − a)ε̄,
0 ≤ a < 1,
with Ȳt ≔ µt + σ̄ W̄t ∈ Rd . The processes (Ȳt )t∈[0,T ] and (St )t∈[0,T ] generate the same filtration. Therefore, I¯TS represents indeed additional information about the terminal stock prices.
(II) Brownian information I¯TW ∈ Rd , where
I¯TW ≔ aσ̄ W̄T + (1 − a)ε̄,
0 ≤ a < 1.
The random variable W̄T is the terminal value of the m-dimensional Brownian motion driving the stock prices.
The last example is a "weaker" form of insider information. Here the investor does not have noisy
information about an FT -measurable random variable, but he has noisy information about the
unknown drift:
(III) Drift information I¯Tµ ∈ Rd , where
I¯Tµ ≔ aµ + (1 − a)ε̄,
0 ≤ a < 1.
In the given model, an insider has the additional information I¯T ∈ {I¯TS , I¯TW , I¯Tµ } at the beginning
of his trading time and I¯T does not change over time. Moreover, the insider information is always
given by a convex combination consisting of the noise and the exact (insider) information. The
latter can either be the terminal stock prices, the terminal value of the driving Brownian motion or
the drift. By considering convex combinations, our model includes an investor without additional
information. We set the scaling factor of the convex combination to a = 0 in order to obtain
the consumption-investment problem with only partial information (without insider information).
A scaling factor of a = 1 corresponds to exact (insider) information without any noise. We
refer to the scaling factor a as the convex factor. In order to guarantee that all considered
consumption-investment problems are well-defined, we do in general not allow the investor
to have exact insider information. Hence, 0 ≤ a < 1. Similar types of insider information
(defined as convex combinations) are considered in the paper of Danilova et al. [2010] in a onedimensional framework and in the paper of Pikovsky & Karatzas [1996] in a multidimensional
framework with complete information. In the latter the components of the noise are assumed to
be independent.
Partial information means here that the investor cannot observe the drift of the stock prices.
However, he knows the constant volatility matrix σ̄. Even though the volatility σ̄ is assumed to
be unknown an investor who can observe the financial market continuously is able to estimate σ̄
perfectly by the quadratic variation of the logarithmic stock prices.
9
3 Financial Market
In this chapter we do a simple transformation of σ̄ W̄t which drives the stock prices. This transformation allows us to write the stock price processes S with a symmetric and non-singular volatility
matrix.
Let us define
Σ := σ̄σ̄ ⊤ .
(3.1)
The following result can for instance be found in Knabner & Barth [2013]. We also give here a
short proof.
Lemma 3.0.1. Let Σ be positive definite. Then there exists a unique symmetric and positive
definite matrix σ ∈ Rd×d such that
Σ = σσ.
The non-singular matrix σ is the so-called positive definite square root of Σ.
Proof. Existence:
Since Σ is symmetric and positive definite, there exists an orthogonal matrix O, i.e. O⊤ = O−1 ,
such that Σ = ODO⊤ , where D is a diagonal matrix D = diag(γ1 , . . . , γd ) and γ1 , . . . , γd are the
positive eigenvalues of Σ. Let us define
1
σ := OD 2 O⊤
1
√
√
with D 2 = diag( γ1 , . . . , γd ). Then we have
1
∈ Rd×d
1
σσ = OD 2 O⊤ OD 2 O⊤ = ODO⊤ = Σ.
(3.2)
Since the eigenvalues of Σ are strictly positive, σ is positive definite and non-singular. The inverse
1
1
matrix is given by σ −1 = OD− 2 O⊤ , where D− 2 = diag( √1γ1 , . . . , √1γ ).
d
Uniqueness:
(i) Let σ ∈ Rd×d be symmetric and positive definite. Let v be the eigenvector of Σ = σσ
√
√
associated to the eigenvalue γ, i.e. σσv − γv = (σ + γId )(σ − γId )v = 0d×1 . Then, (σ −
√
√
γId )v = 0d×1 and v is a eigenvector of σ associated to the eigenvalue γ.
(ii) Let now σ1 and σ2 be positive definite square roots of Σ, i.e. Σ = σ1 σ1 = σ2 σ2 . Moreover,
let vk be the eigenvector of Σ associated to the eigenvalue γk , i.e. Σγk = γk vk , k = 1, . . . , d.
√
Then, Σvk = σ1 σ1 vk = σ2 σ2 vk = γk vk . It follows from (i) that σ1 vk = σ2 vk = γk vk . Hence,
σ1 = σ2 since σ1 and σ2 have the same eigenvalues and eigenvectors.
11
3 Financial Market
Remark 3.0.2. For Σ defined in (3.1) the following statements are equivalent:
(i) rank(σ̄) = d;
(ii) Σ is positive definite.
Theorem 3.0.3. Let σ be the positive definite square root of Σ and let Wt = (Wt1 , . . . , Wtd )⊤ be
defined by
Wt = σ −1 σ̄ W̄t ,
t ∈ [0, T ],
P-a.s.
Then W = (Wt )t∈[0,T ] is a d-dimensional standard Brownian motion with respect to F.
Proof. Wt = σ −1 σ̄ W̄t is measurable w.r.t Ft for all t ∈ [0, T ] as linear combinations of Ft -measurable
random variables. We now apply Lévy’s characterization of Brownian motions (see Theorem 39
in Protter [1990]) in order to show that W is a standard F-Brownian motion:
(i) W is a martingale w.r.t. F since
E[Wt − Ws | Fs ] = σ −1 σ̄ E[W̄t − W̄s | Fs ] = 0d×1 ;
|
{z
}
=0m×1
(ii) The quadratic variation is given by
[W· , W· ]t = [σ −1 σ̄ W̄· , σ −1 σ̄ W̄· ]t = σ −1 σ̄ [W̄· , W̄· ]t σ̄ ⊤ (σ −1 )⊤ = tσ −1 Σσ −1 = tId ,
| {z }
tIm
where Id denotes the d × d−identity matrix.
Due to Theorem 3.0.3 we consider from now on the following dynamics of the price processes S and
B with the non-singular and symmetric volatility matrix σ, and the modified insider information
ITS , ITW , ITλ :
Bt = ert ;
Bond price:
Stock prices:
dSt = diag(St1 , . . . , Std )σ(λdt + dWt ),
S0 = 1d := (1, . . . , 1)⊤ ∈ Rd ,
where for notational reasons we consider λ := σ −1 µ ∈ Rd as the unknown drift (instead
of µ). By virtue of the non-singularity of the volatility matrix σ the unknown drift λ is
well-defined with given initial distribution
Q0 = N (λ0 , Σ0 ) where
λ0 := σ −1 µ0 , Σ0 := σ −1 Σ̄0 σ −1 .
Furthermore, we call
Yt := λt + Wt
the (continuous-time) observation process. The insider information IT ∈ {ITS , ITW , ITλ } is
then defined as follows:
12
ITS := σ −1 I¯TS = aYT + (1 − a)ε;
Stock price information:
Brownian information:
ITW := σ −1 I¯TW = aWT + (1 − a)ε;
ITµ := σ −1 I¯Tµ = aλ + (1 − a)ε,
Drift information:
where the noise ε is given by
d
ε := σ −1 ε̄ = N (0d×1 , Σε )
with
Σε := σ −1 Σ̄ε σ −1 .
Remark 3.0.4. The transformation in Theorem 3.0.3 leads to a loss of information - in the
sense that
σ(Wu : 0 ≤ u ≤ t) = σ(σ̄ W̄u : 0 ≤ u ≤ t) ⊆ σ(W̄u : 0 ≤ u ≤ t).
However, the investor has to take his investment decisions on the basis of the observations of the
stock prices and the insider information. Hence, the loss of information is not relevant
σ(Su : 0 ≤ u ≤ t) = σ(Yu : 0 ≤ u ≤ t) = σ(µu + σ̄ W̄u : 0 ≤ u ≤ t).
We obtain IT out of I¯T by a simple transformation, namely ITS := σ −1 I¯TS . Therefore, it is obvious that I¯T and IT generate the same σ-field. This indicates that IT and I¯T provide the same
information and thus we consider ITS , ITW , ITλ as the given insider information.
13
4 Discrete-Time Consumption-Investment
Problems
In reality, continuously trading is not possible. There are always some time lags where the
investor does not observe the financial market and where he does not trade, for instance, the
investor might reduce his trading frequency in order to relax or to avoid transaction costs. For
this reason, we consider a discrete-time investor who is only able to observe the financial market
and rebalance the portfolio at discrete points in time. Therefore, we approximate the underlying
continuous-time financial market introduced in Chapter 2 and thus consider the price processes
as observed by a discrete-time investor. Without loss of generality we suppose d = m and a
symmetric, positive definite volatility matrix σ (compare Chapter 3). As aforementioned, we call
λ := σ −1 µ ∈ Rd ,
i.e.
µk =
d
X
σkj λj , k = 1, . . . , d,
j=1
the (unknown) drift and we model its uncertainty by assuming λ to be an F0 -measurable random
variable with initial distribution Q0 = N (λ0 , Σ0 ).
4.1 Discretized Financial Market
As in Bäuerle et al. [2012] and in Rogers [2001] we assume the investor to be only able to observe
the stock prices and adjust the portfolio at multiples of an h > 0. We call such an investor an
h-investor or a relaxed investor. The trading time points are then given by tn = nh for
n = 0, 1, . . . , N − 1 with N ∈ N, where we make the convention T = N h. By a direct discretization
of the underlying continuous-time price processes we obtain the discretized financial market:
Bond price at time tn = nh :
Bnh = ernh ;
Stock price at time tn = nh :
k
Snh
= exp
n = 0, . . . , N =
d
P
j=1
T
h,
j
σkj (λj nh + Wnh
) − 12
k = 1, . . . , d.
d
P
j=1
2 nh
σkj
!
,
In this chapter h is always considered as fixed. Therefore, we simplify the notations
Bn
Snk
:= Bnh ,
k
:= Snh
,
n = 0, . . . , N ;
n = 0, . . . , N, k = 1, . . . , d.
k)
Furthermore, we define Rk = (Rn
n∈{1,...,N } for k = 1, . . . , d by


d
d
X
X
1
j
j
2 
k
σkj
h
Rn
:= exp 
σkj (λj h + (Wnh
− W(n−1)h
)) −
2
j=1
j=1
15
4 Discrete-Time Consumption-Investment Problems
such that
Snk =
n
Y
Rik .
i=1
k is the relative price change of stock k at time t = nh. This price change is observed
Here Rn
by the h-investor when he observes the underlying continuous-time financial market at time
tn−1 = (n − 1)h and then again at time tn = nh. The randomness comes from Z = (Zn )n∈{1,...,N } ,
where
j
j
− W(n−1)h
),
Znj ≔ λj h + (Wnh
j = 1, . . . , d.
(4.1)
The logarithmic change in the stock prices is an affine function of Zn = (Zn1 , . . . , Znd )⊤ , i.e.
!
d
d
k
X
Sn+1
1X 2
j
σkj h.
log
=
σ
Z
−
kj n+1
2
Snk
j=1
j=1
For a given stock price Snk the price of stock k at time tn+1 = (n + 1)h is increasing in the weighted
d
P
j
σjk Zn+1
of Zn+1 . The conditional distribution of Zn+1 given the unknown drift λ = θ
sum
j=1
is a multivariate normal distribution
QZ (· | θ) = N (θh, hId ).
(4.2)
Since σ(S1 , . . . , Sn ) = σ(Z1 , . . . , Zn ), we call Z the observation process of the h-investor. Hence,
the information filtration of a regular h-investor is given by FZ = (FnZ )n∈{0,...,N } , where
F0Z
FnZ
:=
:=
{Ω, ∅};
σ(Z1 , . . . , Zn ).
The h-investor with insider information has access to the filtration FIT Z = (FnIT Z )n∈{0,...,N }
which is generated by the observation process Z and the additional information IT
FnIT Z
:=
σ(IT , Z1 , . . . , Zn ) ⊆ Gn := σ(IT , λ, W0 , W1h , . . . , Wnh ),
n = 0, . . . , N.
We refer to an h-investor with insider information also as an h-insider. The filtration G =
(Gn )n∈{0,...,N } is the information filtration of an h-insider who additionally knows the drift.
4.2 Formulation of the Optimization Problem
In this section we formulate the optimization problem for the h-investor in a mathematical way
which we then solve in Section 4.5 for an h-investor with logarithmic and power utility functions.
Consumption-Investment Strategy
At each trading time point tn the h-investor first decides about the consumption rate at which
he consumes until the next trading time point. Afterward, he allocates the remaining wealth
(after consumption) among the bond and the stocks. All decisions are subjected to the constraint that the wealth is non-negative all the time. We call a (2 + d)-dimensional adapted (with
respect to an appropriate filtration) stochastic process π = (cn , bn , an )n∈{0,...,N −1} a (discretetime) consumption-trading strategy, where the quantities have the following meaning
16
4.2 Formulation of the Optimization Problem
• hcn ≥ 0 is the amount of money which is consumed at rate cn ≥ 0 during the time interval
[nh, (n + 1)h);
• bn ∈ R is the number of bonds B in the portfolio during the time interval [nh, (n + 1)h);
• akn ∈ R, k = 1, . . . , d, is the number of stocks S k in the portfolio during the time interval
[nh, (n + 1)h).
Remark 4.2.1. A negative number of the bond B means that the investor takes out a credit loan.
A negative number of the stock S k means that the investor short sells stock k.
The value Xn of the portfolio π at time tn = nh consists of the wealth invested into the stocks
S k , k = 1, . . . , d, and the wealth invested into the bond B, i.e.
Xn := a⊤
n−1 Sn + bn−1 Bn .
(4.3)
We call Xn the (total) wealth. The random variable Xn is the wealth at time tn = nh immediately
before consuming and trading. Then
Xn − hcn
is the remaining wealth at time tn = nh (after consumption) which is invested into the stocks and
the bond. For the following definition we refer to Korn & Korn [2001].
Definition 4.2.2. A (discrete-time) consumption-trading strategy π = (cn , bn , an )n∈{0,...,N −1} is
called self-financing if the corresponding wealth process satisfies
Xn − hcn = a⊤
n Sn + bn Bn
P−a.s.
for n = 0, . . . , N − 1.
(4.4)
Self-financing means intuitively that the remaining wealth after consumption is completely reallocated among the assets and there is no inflow of money. In this thesis, we restrict ourselves to
self-financing strategies.
k , k = 1, . . . , d, and β by
For n = 0, . . . , N − 1 we define αn
n
k
:=
αn
βn :=
k
ak
n Sn
Xn −hcn
b n Bn
Xn −hcn
k
with αn
:= 0
with βn := 0
if Xn − hcn = 0;
(4.5)
if Xn − hcn = 0.
The self-financing condition (4.4) can then equivalently be formulated as
⊤
βn = (1 − αn
1d ),
where 1d := (1, . . . , 1)⊤ ∈ Rd . It follows that the strategy π is fully described by cn and αn . Thus,
we call the (1 + d)-dimensional process π = (cn , αn )n∈{0,...,N −1} a (discrete-time) consumptioninvestment strategy, where
hcn
⊤1 )
(1 − αn
d
k , k = 1, . . . , d,
αn
is the amount of money which is consumed at rate cn during
the time interval [nh, (n + 1)h);
is the fraction of wealth which is invested into the bond at time tn = nh;
is the fraction of wealth which is invested into stock k at time tn = nh.
17
4 Discrete-Time Consumption-Investment Problems
Wealth Process
Using the self-financing condition we obtain an important recursion for the wealth process (4.3)
Xn =a⊤
n−1 Sn + bn−1 Bn =
d
X
k=1
k
akn−1 Sn−1
Bn
Snk
+ bn−1 Bn−1
.
k
Bn−1
Sn−1
Using the transformation (4.5) and the self-financing condition (4.4), we obtain
!
d
d
X
X
k
k
k
Xn =
αn−1 (Xn−1 − hcn−1 )Rn + 1 −
αn−1 (Xn−1 − hcn−1 )erh
k=1
=(Xn−1 − hcn−1 )e
rh
k=1
⊤
−rh
(1 + αn−1 (Rn e
− 1d )).
k , k = 1, . . . , d, depend on the
In order to emphasize that the relative price changes of the stocks Rn
k
observation Zn , we write R (Zn ), i.e.


d
d
X
X
1
2 
Rk (Zn ) := exp 
σkj Znj −
σkj
h
2
j=1
j=1
with R(Zn ) = (R1 (Zn ), . . . , Rd (Zn ))⊤ ∈ Rd . Altogether, we obtain the so-called wealth recursion:
X0π
Xnπ
≔
x0 ;
=
rh
e
(4.6)
π
⊤
(Xn−1
− hcn−1 )(1 + αn−1
(e−rh R(Zn ) − 1d )),
n = 1, . . . , N.
We write Xnπ in order to make clear that this wealth is obtained under the consumptioninvestment strategy π.
Remark 4.2.3. In the terminal wealth problem, the aim is to maximize only the utility of the
terminal wealth. We then call the consumption-investment strategy π = (αn )n∈{0,...,N −1} with
a consumption rate cn ≡ 0 an investment or portfolio strategy. The corresponding wealth
recursion is given by
π
⊤
Xnπ = Xn−1
erh (1 + αn−1
(e−rh R(Zn ) − 1d )).
Optimization Problem and Admissible Consumption-Investment Strategy
Now, we are able to formulate precisely the h-insider’s optimization problem which we denote
by (P h ). The h-insider aims to maximize the expected utility of terminal wealth, evaluated by
the utility function Up , and the intermediate consumption, evaluated by the utility function Uc ,
given the additional information IT = iT with IT ∈ {ITW , ITS , ITλ }. We maximize the expected
utility over all admissible consumption-investment strategies and under the constraint that the
wealth is non-negative all the time. Hence, (P h ) can be written as follows
 N
P

π )| I = i

E
hU
(c
)
+
U
(X

c
n−1
p
T
T → max;
N


 n=1
(P h ) Xnπ ≥ 0, n = 0, . . . , N ;


X0π = x0 ;



π = (cn , αn )n∈{0,...,N −1} admissible consumption-investment strategy.
18
4.2 Formulation of the Optimization Problem
The investor can take his decisions only on the basis of his information which are given by the
observations (of the stock prices) and the insider information. Therefore, an admissible
consumption-investment strategy has to be adapted with respect to the filtration FIT Z . Due to the
normal distributions of Z and IT (see also Lemma 4.4.2), we can only guarantee a non-negative
wealth (except for exact stock price information; see Remark 4.4.10) or in other words, we can
only avoid the risk of going bankrupt, if cn and αn satisfy
αn
hcn
D := {α ∈ [0, 1]d : α⊤ 1d ≤ 1},
∈
∈ [0, x] with x being the current level of wealth
for all n = 0, . . . , N − 1. By virtue of the convention U (x) := −∞, x < dom(U ), it is obvious that
choosing an investment decision αn , which is not in the convex set D, or consuming more than
the current wealth, will lead to an expected utility of −∞. Thus, for the h-investor it is never
optimal to short sell a stock in order to buy a bond or to take out a loan in order to buy stocks
(compare also Bäuerle et al. [2012]). Therefore, it is enough to consider only strategies as defined
below.
Definition 4.2.4. An admissible consumption-investment strategy is an FIT Z -adapted
stochastic process π = (cn , αn )n∈{0,...,N −1} with αn ∈ D and cn h ∈ [0, x], where x is the current
level of wealth. We denote by Ahn the set of admissible consumption-investment strategies when
starting at time tn = nh.
If we define
∗
V0π
(x0 , iT ) :=
E
"
N
X
n=1
π
hUc (cn−1 ) + Up (XN
)|
X0π
= x0 , IT = iT
#
for a strategy π, then the value of (P h ) is given by
∗
(x0 , iT ).
V0∗ (x0 , iT ) := sup V0π
π∈Ah
0
We call a consumption-investment strategy π ∗ ∈ Ah0 optimal for the consumption-investment
problem (P h ) if
∗
∗
V0π
∗ (x0 , iT ) = V0 (x0 , iT )
for any iT ∈ Rd , x0 ∈ R>0 .
Remark 4.2.5. Note that the wealth depends on the unknown drift. Unlike classical consumptioninvestment problems, we (implicitly) have an additional expectation with respect to the initial
distribution of λ updated by the insider information. If we denote this conditional distribution by
Q0 (· | iT ), then we can equivalently write
#
Z "X
N
∗
π
π
V0π (x0 , iT ) = E
hUc (cn−1 ) + Up (XN ) | X0 = x0 , λ = θ, IT = iT Q0 (dθ | iT ).
n=1
π
Since the wealth Xn+1
depends on Zn+1 and an admissible strategy is FIT Z -adapted, we are
interested in the conditional distribution of Zn+1 given the information FnIT Z = σ(IT , Z1 , . . . , Zn ).
The aim of Section 4.3 is to determine this conditional distribution (see Definition 4.3.21).
19
4 Discrete-Time Consumption-Investment Problems
Approach
For an admissible consumption-investment strategy π = (cn , αn )n∈{0,...,N −1} we consider the
wealth recursion (4.6). The wealth depends on the unknown drift through the distribution of the
observation Z, namely QZ (· | θ). However, cn and αn are FnIT Z -measurable and consequently do
not depend (explicitly) on the unknown drift. Moreover, due to FnZ ⊆ FnIT Z , the set of admissible
consumption-investment strategies of the h-insider is greater than the set of admissible strategies
of a regular h-investor, but so far, the insider information is not included in the distribution of Z.
Hence, (P h ) is not a standard consumption-investment problem and it is not immediately clear
how to solve it. Nevertheless, by an enlargement and filtering step, we are able to formulate (P h )
as a (standard) Markov Decision Process (MDP) with a state space that contains the insider
information.
In order to solve (P h ) we follow the approach as described below:
(Step 1) In standard portfolio optimization problems the state of an investor is described by
the wealth. In order to incorporate the insider information, we enlarge the state space
such that the latter contains the insider information. We then look for the conditional
distribution of Zn+1 given the unknown drift and the insider information. In (Step 1),
the enlargement step, the drift is assumed to be known;
(Step 2) In the second step we incorporate the partial information into the optimization problem
obtained after (Step 1). Consequently, we estimate the unknown drift on the basis of
the insider information and the observations of the stock prices, i.e. we look for the
posterior distribution of the drift. (Step 2) is the filtering step;
(Step 3) (Step 1) and (Step 2) result in the conditional distribution of Zn+1 given the available information σ(IT , Z1 , . . . , Zn ). In the last step we then formulate the consumptioninvestment problem with partial and insider information as an MDP (with an enlarged
state space) and solve it recursively via the Bellman equation.
4.3 Enlargement and Filtering
In this section we reduce the optimization problem (P h ) with partial and insider information to
a standard MDP. In order to simplify the notation we introduce some abbreviations
Θ
:= Rd is the parameter space.
The unknown drift λ takes its values in Θ;
Z
:= Rd is the observation space.
The observations Zn , n = 1, . . . , N, take their values in Z;
I
:= Rd is the space of the insider information.
The insider information IT takes its values in I.
We endow these spaces with the Borel σ-field. We denote the Borel σ-field on Rd by B(Rd ). For
the observations Zn , n = 1, . . . , N, we introduce the following notation
Hn :=
hn :=
20
(Z1 , . . . , Zn ) ∈ Z n
(z1 , . . . , zn )
are the past observations at time tn = nh.
We call Hn also the history;
is a realization of Hn , where Hn = hn means Z1 = z1 , . . . , Zn = zn .
4.3 Enlargement and Filtering
Action: (cn , αn )
−rh
((xn −hcn )erh (1+α⊤
R(zn+1 )−1d)), iT )
n (e
State at time
tn = nh :
(xn , iT )
with a f ixed
h>0
Random transition
I
State at time
tn+1 = (n + 1)h :
(xn+1 , iT )
Z
QnT (· | θ, iT , z1 , . . . , zn )
hUc (cn )
Figure 4.1: Evolution of the state consisting of the wealth and the insider information, where
the transition to the next state depends on the unknown drift and the history
Furthermore, we make use of the abbreviation
Kn := a2 (N − n)hId + (1 − a)2 Σε ,
n = 0, . . . , N.
Kn depends on the convex factor a ∈ [0, 1) and on the time lag h > 0.
From the outset, we consider, as in standard consumption-investment problems, the wealth process (Xnπ )n∈{0,...,N } as the (observable) state process of the investor with state space dom(Up ).
4.3.1 Enlargement
In the enlargement step, we only incorporate the insider information (not the partial information).
Hence, we consider the unknown drift λ = θ as given.
At the beginning of the trading time, the investor has a given initial wealth x0 and in addition he
knows the value iT of the insider information IT . Therefore, we include the insider information
by enlarging the investor’s state space by the space of the insider information. We consequently
consider (Xnπ , IT ) as the state process of the h-investor. The evolution of the state process can
be described as follows:
Let Xnπ = x be the wealth at time tn = nh. Then the current state is given by the pair
(x, iT ) consisting of the current wealth and the insider information IT = iT . Once an action
(cn , αn ) is chosen, where π = (cn , αn )n∈{0,...,N −1} is an admissible strategy, there is a random
transition to the next state. Whereas the insider information does not change over time, the
wealth at time tn+1 = (n + 1)h is determined by the wealth recursion. Hence, the state at
time tn+1 = (n + 1)h is given by
⊤ −rh
(x − hcn )erh (1 + αn
(e
R(Zn+1 ) − 1d )), iT .
(4.7)
This (future) state depends on the (future) observation Zn+1 and is therefore random. Since
π is an admissible strategy and consequently FIT Z -adapted, the action (cn , αn ) is a function of
the insider information IT = iT and the past observations Hn = hn . In order to determine the
distribution of the state at time tn+1 , we look for the conditional distribution of Zn+1 given the
insider information IT = iT , the history Hn = hn and the drift λ = θ. We denote this conditional
21
4 Discrete-Time Consumption-Investment Problems
distribution by QInT Z (· | θ, iT , hn ). Figure 4.1 illustrates the evolution of the state process (compare
also Figure 5.1 in Bäuerle & Rieder [2011]).
In the following Subsections (I) to (III) we determine QInT Z for stock price, Brownian and drift
information.
Remark 4.3.1. The following results depend on the type of insider information. However, for
notational reasons, we use in general the same notations as long as it is clear from the context
which type of insider information we consider. Furthermore, if we do not explicitly specify the
type of insider information, then the corresponding results hold for stock price, Brownian and
drift information.
(I) Stock Price Information
Firstly, we consider stock price information, i.e. IT = ITS .
Theorem 4.3.2. For n = 0, . . . , N − 1, the (regular) conditional distribution QInT Z of Zn+1 given
the unknown drift λ = θ, the stock price information ITS = iT and the past observations Hn = hn
follows a multivariate normal distribution
!
!
n
X
−1
2
2
IT Z
−1
zk , hKn Kn+1
(1 − a) Σε θ + aiT − a
Qn (· | θ, iT , hn ) = N hKn
k=1
with
QI0T Z (·
|
θ, iT , h0 ) := QI0T Z (·
| θ, iT )
= N (hK0−1 ((1 − a)2 Σε θ + aiT ), hK0−1 K1 ).
Remark 4.3.3. It holds that
P(Zn+1 ∈ ·
| λ, ITS , Hn )
hKn−1
=N
2
(1 − a)
Σε λ + aITS
2
−a
n
X
k=1
Zk
!
, hKn−1 Kn+1
!
a.s.
with P(Zn+1 ∈ · | λ, ITS , Hn ) = QInT Z (· | θ, iT , hn ) on the set {λ = θ, ITS = iT , Hn = hn } (see e.g.
Breiman [1993]). We therefore also use the notation QInT Z (· | λ, ITS , Hn ) for P(Zn+1 ∈ · | λ, ITS , Hn ).
Proof. Firstly, we determine the conditional distribution of the insider information given the
unknown drift and the past observations. By Theorem 8.37 in Klenke [2014] there exists a regular
conditional distribution of ITS given λ and Hn . Hence, the conditional characteristic function can
be used in order to determine its conditional distribution (see e.g. Lemma 6.13 Chapter 2 in
Karatzas and Shreve [1991]). Let p ∈ Rd . Then
i
h
E exp ip⊤ (a(T λ + WT ) + (1 − a)ε) | λ, Hn
"
!!
#
N
X
⊤
=E exp ip
aT λ + a
(Wkh − W(k−1)h ) + (1 − a)ε
| λ, Hn
k=1



=E exp ip⊤ a(T − nh)λ + a
k=1
= exp ip⊤ a(T − nh)λ + a
Zk
= exp ip⊤ a(T − nh)λ + a
22
n
X
n
X
k=1
n
X
k=1
Zk + a
k=n+1
!! 
!
N
X



(Wkh − W(k−1)h ) + (1 − a)ε | λ, Hn 

E exp ip⊤ a
!
1
Zk − p⊤ Kn p ,
2
N
X
k=n+1
 
(Wkh − W(k−1)h ) + (1 − a)ε
4.3 Enlargement and Filtering
where we use the independence of ε, W and λ. Hence,
P(ITS ∈ · | λ, Hn ) = N
a(T − nh)λ + a
n
X
Zk , Kn
k=1
!
a.s.
with P(ITS ∈ · | λ, Hn ) = P(ITS ∈ · | λ = θ, Hn = hn ) on the set {λ = θ, Hn = hn }. Furthermore, given
the unknown drift, the observations Zk , k = 1, . . . , n + 1, are conditionally independent. Thus,
P(Zn+1 ∈ · | λ, Hn ) = P(Zn+1 ∈ · | λ) = N (hλ, hId )
a.s.
There also exists a (regular) conditional distribution of Zn+1 given λ and Hn . Denoting by fX (· | y)
the conditional density function of a random variable X given a random variable Y = y, we obtain
by applying the extension of Bayes’ theorem for absolutely continuous random variables (see e.g.
DeGroot [1970])
fZn+1 (zn+1 | θ, iT , hn )
=
fI S (iT | θ, hn , zn+1 )fZn+1 (zn+1 | θ, hn )
T
fI S (iT | θ, hn )
T
det Kn
1
⊤
=
(z
−
hθ)
(z
−
hθ)
exp
−
n+1
n+1
d p
2h
(2π) 2 h det Kn+1

!!
!!⊤
n
n
X
X
1

× exp 
zk
Kn−1 iT − a(T − nh)θ + a
zk
iT − a(T − nh)θ + a
2
√
k=1
k=1
× exp
−
1
2
iT − a(T − (n + 1)h)θ + a
−1
× Kn+1
n
X
zk + azn+1
k=1
!!⊤
n
X
iT − a(T − (n + 1)h)θ + a
zk + azn+1
k=1
!! !
.
After rearranging the exponent, we see that the (regular) conditional distribution of Zn+1 given
λ, ITS and Hn has density
fZn+1 (zn+1 | θ, iT , hn )

√
1
det Kn
exp −
=
d p
2h
(2π) 2 h det Kn+1
zn+1 −
−1
×Kn+1
Kn
hKn−1
zn+1 −
2
2
(1 − a) Σε θ + aiT − a
hKn−1
2
n
X
zk
k=1
!!!⊤
2
(1 − a) Σε θ + aiT − a
n
X
k=1
zk
!!!!
.
(II) Brownian Information
We assume the investor to have Brownian information, i.e. IT = ITW .
Theorem 4.3.4. For n = 0, . . . , N − 1, the (regular) conditional distribution QInT Z of Zn+1 given
the unknown drift λ = θ, the Brownian information ITW = iT , and the past observations Hn = hn
23
4 Discrete-Time Consumption-Investment Problems
follows a multivariate normal distribution
QInT Z (· | θ, iT , hn ) = N
hKn−1 K0 θ + aiT − a2
n
X
!
zk , hKn−1 Kn+1
k=1
!
with QI0T Z (· | θ, iT , h0 ) := QI0T Z (· | θ, iT ) = N (hK0−1 (K0 θ + aiT ), hK0−1 K1 ).
Proof. The proof follows the same lines as the proof of Theorem 4.3.2. We determine again via the
characteristic function the conditional distribution of the insider information given the unknown
drift and the past observations. Let p ∈ Rd . Then
"
!!
#
N
h
i
X
⊤
⊤
E exp ip (aWT + (1 − a)ε) | λ, Hn =E exp ip
a
(Wkh − W(k−1)h ) + (1 − a)ε
| λ, Hn
k=1
= exp ip
i.e.
P(ITW
∈ · | λ, Hn ) = N −anhλ + a
n
P
⊤
Zk , Kn
k=1
−anhλ + a
n
X
Zk
k=1
!
!
1 ⊤
− p Kn p ,
2
a.s. Using the same notation for the condi-
tional density as introduced in the proof before, we obtain by applying the extension of Bayes’
theorem
fZn+1 (zn+1 | θ, iT , hn )

√
1
det Kn
exp −
=
d p
2h
(2π) 2 h det Kn+1
zn+1 − (hKn−1 (K0 θ + aiT
−1
×Kn+1
Kn
2
−a
n
X
k=1
!⊤
zk ))
zn+1 − (hKn−1 (K0 θ + aiT
2
−a
n
X
k=1
!!
zk ))
.
Remark 4.3.5. Since σ(λ, ITS , Hn ) = σ(λ, ITW , Hn ) Theorem 4.3.4 follows directly from Theorem
4.3.2 using Remark 4.3.3 and ITW = ITS − aT λ.
(III) Drift Information
Eventually, let the investor have drift information, i.e. IT = ITλ .
Theorem 4.3.6. For n = 0, . . . , N − 1, the (regular) conditional distribution QInT Z of Zn+1 given
the unknown drift λ = θ, the insider information ITλ = iT and the past observations Hn = hn
follows a multivariate normal distribution
QInT Z (· | θ, iT , hn ) = N (hθ, hId )
with QI0T Z (· | θ, iT , h0 ) := QI0T Z (· | θ, iT ) = N (hθ, hId ).
Remark 4.3.7. For an investor with drift information who additionally knows the drift, the
insider information does not provide (relevant) additional information.
24
4.3 Enlargement and Filtering
Proof. We first determine via the characteristic function the conditional distribution of the insider
information given the unknown drift and the past observations. Let p ∈ Rd . Then
h
i
1
2 ⊤
⊤
⊤
E exp ip (aλ + (1 − a)ε) | λ, Hn
= exp ip λ − (1 − a) p Σε p ,
2
i.e. P(ITλ ∈ · | λ, Hn ) = N aλ, (1 − a)2 Σε a.s. By applying the extension of Bayes’ Theorem, we
obtain for the conditional density of Zn+1 given the unknown drift λ = θ, the drift information
ITλ = iT , and the history Hn = hn
fZn+1 (zn+1 | θ, iT , hn ) = fZn+1 (zn+1 | θ).
For n = 0, . . . , N we introduce the following abbreviation for the sum of the observations Z1 , . . . , Zn
Yn :=
n
X
k=1
Zk
(with Y0 ≡ 0).
The random variable Yn takes its values in Z. Furthermore, we denote by yn :=
tion of Yn . We refer to the sum Yn also as the observation.
n
P
zk a realiza-
k=1
As seen in Theorem 4.3.2, Theorem 4.3.4 and Theorem 4.3.6, (QInT Z ) depends on the past observations, z1 , . . . , zn , only through the number n and the sum of the observations yn . Thus, it
follows:
Corollary 4.3.8. For n = 1, . . . , N − 1 the sum Yn =
n
P
Zk is a sufficient statistic for (QInT Z ),
k=1
i.e. there exists a sequence of transition kernels (QZ
n ) from Θ × I × Z to Z such that
QInT Z (C | θ, iT , hn ) = QZ
n (C | θ, iT ,
n
X
zk )
k=1
for all C ∈ B(Z), θ ∈ Θ, iT ∈ I and hn ∈ Z n . Moreover, QZ
n (· | θ, iT , yn ) has a Lebesgue density
which we denote by
qnZ (zn+1 | θ, iT , yn ).
To simplify the notation, we use for zn+1 ∈ Rd the short notation qnZ (zn+1 | θ, iT , yn )dzn+1 (in1
d
stead of qnZ (zn+1 | θ, iT , yn )dzn+1
. . . zn+1
).
Remark 4.3.9. For stock price and Brownian information, it is a necessary condition for the
Z
Z
existence of qN
−1 that exact insider information is excluded. Otherwise QN −1 has a singular co−1
variance matrix, namely hKN −1 KN = 0d×d . This means that for an insider with exact Brownian
or stock price information who additionally knows the drift the last decision at time t = (N − 1)h
is deterministic.
25
4 Discrete-Time Consumption-Investment Problems
Action: (cn , αn )
State at time
tn = nh :
(xn , iT , yn )
f or a f ixed
h>0
−rh
((xn −hcn )erh (1+α⊤
R(zn+1 )−1d)), iT , yn + zn+1 )
n (e
Random transition
State at time
tn+1 = (n + 1)h :
(xn+1 , iT , yn +zn+1 )
Q̂Z
n (· | iT , yn )
hUc (cn )
Figure 4.2: Evolution of the state consisting of the wealth, the insider information and the sum
of observations, where the transition to the next state depends only on the current
state
The aim is to formulate the consumption-investment problem (of the h-investor) with partial and
insider information as an MDP. An important feature of a (standard) MDP is the Markovian
structure of the evolution of the state. However, the distribution QZ
n (which determines the distribution of the next state) depends through the sum of the observations on the history. Thus, we
enlarge the state space by the space of the observation Z, i.e. we include the sufficient statistic
n
P
Yn =
Zk (observation) into the state, and consider (Xnπ , IT , Yn ) as the state process of the
k=1
h-investor. The evolution of the state process can be described as follows:
Let Xnπ = x be the wealth and Yn = y the observation at time tn = nh. Then the current
state is given by the triplet (x, iT , y) consisting of the current wealth, the insider information
and the current observation. Once chosen an admissible action (cn , αn ), there is a random
transition to the next state. Contrary to (4.7), we additionally update the current observation by the (future) observation Zn+1 . Consequently, the state at time tn+1 = (n + 1)h is
given by
⊤ −rh
(x − hcn )erh (1 + αn
(e
R(Zn+1 ) − 1d )), iT , y + Zn+1 .
The observation Zn+1 is random, where its conditional distribution QZ
n depends through (iT , y)
on the current state. However, QZ
n is independent of the past.
4.3.2 Filtering
In the filtering step we incorporate the partial information. Unlike the previous subsection, we do
not assume the unknown drift λ to be given. The distribution of the observation Zn , n = 1, . . . , N,
depends on the unknown drift so that the state process (Xnπ , IT , Yn ) can be considered as a Partially Observable MDP as treated in Bäuerle & Rieder [2011].
In order to convert the Partially Observable MDP into a (standard) MDP, we look for the conditional distribution of the unknown λ given the available information Hn = hn and IT = iT
(posterior distribution of λ), i.e. we want to estimate the unknown drift on the basis of the observations and the insider information. For this purpose, we define the Bayes operator analogously
to Bäuerle & Rieder [2011].
26
4.3 Enlargement and Filtering
Definition 4.3.10. The Bayes operator Φn : I × Z × P(Θ) × Z → P(Θ) is defined by
R Z
qn (z|θ,iT ,y)ρ(dθ)


 CR
if denominator > 0
Z (z|θ,i ,y)ρ(dθ)
qn
T
Φn+1 (iT , y, ρ, z)(C) ≔


Θ
ρ(C)
if denominator = 0,
where C ∈ B(Θ), n = 0, . . . , N − 1. P(Θ) is the space of all probability measures on Θ.
Using the Bayes operator, we define the filter recursion by which we can compute recursively
the conditional distribution of the unknown drift given the available information (observations
and insider information). The latter is often called posterior distribution of the unknown drift.
Compared to the filter equation given by Bäuerle & Rieder [2011], we choose as initial distribution
the given distribution Q0 (·) but updated by the insider information.
Definition 4.3.11. Let Q0 (· | iT ) := P(λ ∈ · | IT = iT ) and let yn =
z1 , . . . , zn+1 . Then we define the filter recursion
µ0 (C | iT ) ≔
µn+1 (C | iT , hn , zn+1 ) ≔
Q0 (C | iT ),
n
P
zk for observations
k=1
C ∈ B(Θ)
Φn+1 (iT , yn , µn (· | iT , z1 , . . . , zn ), zn+1 )(C),
n = 0, 1, . . . , N − 2.
We use the following lemma in order to show that µn is a conditional distribution of the unknown
drift given the available information.
Lemma 4.3.12. Let v : I × Z n × Θ → R be such that all following integrals exist. Then for all
(iT , hn−1 ) ∈ I × Z n−1 with hn−1 = (z1 , . . . , zn−1 ) it holds that
"
TZ
v(iT , hn−1 , zn , θ)QIn−1
(dzn | θ, iT , hn−1 )µn−1 (dθ | iT , hn−1 )
(4.8)
=
$
TZ
v(iT , hn−1 , zn , θ)µn (dθ | iT , hn−1 , zn )QIn−1
(dzn | θ′ , iT , hn−1 )µn−1 (dθ′ | iT , hn−1 ) (4.9)
for n = 1, . . . , N − 1.
Proof. Let yn−1 be the sum of n − 1 observations, i.e. yn−1 =
n−1
P
zk . Firstly, we consider (4.8).
k=1
By Corollary 4.3.8 we obtain
"
TZ
v(iT , hn−1 , zn , θ)QIn−1
(dzn | θ, iT , hn−1 )µn−1 (dθ | iT , hn−1 )
=
"
Z
v(iT , hn−1 , zn , θ)qn−1
(zn | θ, iT , yn−1 )dzn µn−1 (dθ | iT , hn−1 ).
On the other hand, for (4.9) we obtain by Corollary 4.3.8 and by the filter recursion
$
TZ
v(iT , hn−1 , zn , θ)µn (dθ | iT , hn−1 , zn )QIn−1
(dzn | θ′ , iT , hn−1 )µn−1 (dθ′ | iT , hn−1 )
=
$
Z
v(iT , hn−1 , zn , θ)µn (dθ | iT , hn−1 , zn )qn−1
(zn | θ′ , iT , yn−1 )dzn µn−1 (dθ′ | iT , hn−1 )
27
4 Discrete-Time Consumption-Investment Problems
=
$
v(iT , hn−1 , zn , θ)
Z
× Φn (iT , yn−1 , µn−1 , zn )(dθ)qn−1
(zn | θ′ , iT , yn−1 )dzn µn−1 (dθ′ | iT , hn−1 )
$
=
v(iT , hn−1 , zn , θ)
=
×R
"
Z (z | θ, i , y
qn−1
n
T n−1 )µn−1 (dθ | iT , hn−1 )
q Z (zn | θ′ , iT , yn−1 )dzn µn−1 (dθ′ | iT , hn−1 )
Z
′′
qn−1 (zn | θ , iT , yn−1 )µn−1 (dθ′′ | iT , hn−1 ) n−1
Z
v(iT , hn−1 , zn , θ)qn−1
(zn | θ, iT , yn−1 )
R Z
q
(zn | θ′ , iT , yn−1 )µn−1 (dθ′ | iT , hn−1 )
µn−1 (dθ | iT , hn−1 )dzn
× R Zn−1
qn−1 (zn | θ′′ , iT , yn−1 )µn−1 (dθ′′ | iT , yn−1 )
"
Z
=
v(iT , hn−1 , zn , θ)qn−1
(zn | θ, iT , yn−1 )µn−1 (dθ | iT , hn−1 )dzn
=
"
Z
v(iT , hn−1 , zn , θ)qn−1
(zn | θ, iT , yn−1 )dzn µn−1 (dθ | iT , hn−1 ).
We use here Fubini’s theorem to interchange the order of the integrals.
Theorem 4.3.13. For n = 0, . . . , N −1 we have that µn (· | IT , Z1 , . . . , Zn ) is a (version of the) conditional distribution of the unknown drift given the insider information and the past observations,
i.e.
µn (C | IT , Z1 , . . . , Zn ) = P(λ ∈ C | IT , Z1 , . . . , Zn ),
C ∈ B(Θ).
Proof. In order to prove that µn is a conditional distribution of λ given σ(IT , Hn ) we show
(i) E[1Cn µn (C | IT , Hn )] = E[1Cn 1C (λ)] for all Cn ∈ σ(IT , Hn ), n = 1, . . . , N − 1, with
C0 ∈ σ(IT ). Note that Cn ∈ σ(IT , Hn ) means Cn = [(IT , Hn ) ∈ Bn ] for some Bn ∈
B(I × Z n ) (Borel σ-field over Rd(n+1) );
(ii) µn (C | IT , Hn ) is measurable w.r.t. σ(IT , Hn ) for any C ∈ B(Θ).
We start with (i).
(i) We show (i) by induction.
Initial step:
n = 0 : X since: by definition µ0 (C | IT ) := Q0 (C | IT ) = P(λ ∈ C | IT );
n = 1 : X since (note that y0 ≡ 0):
28
4.3 Enlargement and Filtering
Using the initial step for n = 0 and Lemma 4.3.12 we obtain
E[µ1 (C | IT , Z1 )1C1 ]
=E[E[E[µ1 (C | IT , Z1 )1C1 | λ, IT ] | IT ]]
Z
′
′
1C (θ )µ1 (dθ | IT , Z1 )1C1 | λ, IT | IT
=E E E
"
′
′
Z
1C (θ )µ1 (dθ | IT , z1 )1B1 (IT , z1 )Q0 (dz1 | λ, IT , y0 ) | IT
=E E
$
′
′
Z
1C (θ )1B1 (IT , z1 )µ1 (dθ | IT , z1 )Q0 (dz1 | θ, IT , y0 )Q0 (dθ | IT )
=E
$
′
′
Z
1C (θ )1B1 (IT , z1 )µ1 (dθ | IT , z1 )Q0 (dz1 | θ, IT , y0 )µ0 (dθ | IT )
=E
"
Z
1C (θ)1B1 (IT , z1 )Q0 (dz1 | θ, IT , y0 )µ0 (dθ | IT )
=E
=E[1C (λ)1C1 ].
Inductive step: Let µn (· | IT , Hn ) = P(λ ∈ · | IT , Hn ) (induction hypothesis).
n
P
n → n + 1 : Let Yn :=
Zk . Then
k=1
E[1Cn+1 µn+1 (C | IT , Hn , Zn+1 )]
=E E E 1Cn+1 µn+1 (C | IT , Hn , Zn+1 ) | λ, IT , Hn | IT , Hn
Z
′
′
=E E E 1Cn+1 1C (θ )µn+1 (dθ | IT , Hn , Zn+1 ) | λ, IT , Hn | IT , Hn
Z
1Bn+1 (IT , Hn , zn+1 )
=E E
Z
′
′
Z
× 1C (θ )µn+1 (dθ | IT , Hn , zn+1 )Qn (dzn+1 | λ, IT , Yn ) | IT , Hn .
Now, we first use the induction hypothesis and then apply Lemma 4.3.12. Consequently
E[1Cn+1 µn+1 (C | IT , Hn , Zn+1 )]
"$
=E
1Bn+1 (IT , Hn , zn+1 )
′
′
× 1C (θ )µn+1 (dθ |
=E
"
IT , Hn , zn+1 )QZ
n (dzn+1
1Bn+1 (IT , Hn , zn+1 )1C (θ)QZ
n (dzn+1
=E 1Cn+1 1C (λ) .
#
| θ, IT , Yn )µn (dθ | IT , Hn )
| θ, IT , Yn )µn (dθ | IT , Hn )
(ii) The measurability of µn w.r.t. σ(IT , Hn ) can easily be seen from the explicit representations
of µn in Theorem 4.3.15, Theorem 4.3.17 and Theorem 4.3.19.
From (i) and (ii) it follows that µn (C | IT , Hn ) = E[1C (λ) | IT , Hn ] a.s., i.e. µn (· | IT , Hn ) is a
version of the conditional distribution of λ given IT , Hn .
29
4 Discrete-Time Consumption-Investment Problems
In the following subsections (I) to (III) we first determine the initial distribution Q0 (· | iT ). By
using Q0 (· | iT ), we are able to apply the filter recursion in order to derive the conditional
distribution of the unknown drift µn (· | IT , Z1 , . . . , Zn ) for all three types of insider information.
(I) Stock Price Information
Let the investor have stock price information, i.e. IT = ITS .
Lemma 4.3.14. The initial distribution Q0 (· | iT ) in the filtering recursion (with stock price
information) follows a multivariate normal distribution
−1
−1 Q0 (· | iT ) = N λ0 + aT Σ0 K0 + a2 T 2 Σ0
(iT − aλ0 T ) , K0 K0 + a2 T 2 Σ0
Σ0 .
Proof. We consider
φS
:=
(λ − λ0 ) − aT Σ0 K0 + a2 T 2 Σ0
ITS
=
a (λT + WT ) + (1 − a)ε.
−1 ITS − aλ0 T ,
We show that φS and ITS are independent:
• φS and ITS are uncorrelated since
−1
Cov(λ, ITS ) − aT Σ0 K0 + a2 T 2 Σ0
Cov(ITS , ITS )
−1
= aT Σ0 − aT Σ0 K0 + a2 T 2 Σ0
a2 T 2 Σ0 + a2 T Id + (1 − a)2 Σε
= 0d×d ;
Cov(φS , ITS )
=
⊤ ⊤
2d with m , m ∈ Rd . By the indepen• (φS , ITS ) is Gaussian since: Let m = (m⊤
1
2
1 , m2 ) ∈ R
dence of W, λ and ε we obtain
S ⊤ φ
E exp im
ITS
2
2 2
−1
= exp(im⊤
)λ0 )
1 (−Id + (aT ) Σ0 (K0 + a T Σ0 )
2 2
−1 S
S
× E[exp(im⊤
IT ) + im⊤
1 (λ − aT Σ0 (K0 + a T Σ0 )
2 IT )]
2
2 2
−1
= exp(im⊤
)λ0 )
1 (−Id + (aT ) Σ0 (K0 + a T Σ0 )
× E[exp(i(m1 − (aT )2 (K0 + a2 T 2 Σ0 )−1 Σ0 m1 + (aT )m2 )⊤ λ)]
× E[exp(i(−a2 T (K0 + a2 T 2 Σ0 )−1 Σ0 m1 + am2 )⊤ WT )]
× E[exp(i(−(1 − a)aT (K0 + a2 T 2 Σ0 )−1 Σ0 m1 + (1 − a)m2 )⊤ ε)]
⊤
= eim1 (−Id +(aT )
2
Σ0 (K0 +a2 T 2 Σ0 )−1 )λ0 i(m1 −(aT )2 (K0 +a2 T 2 Σ0 )−1 Σ0 m1 +aT m2 )⊤ λ0
e
− 12 (m1 −(aT )2 (K0 +a2 T 2 Σ0 )−1 Σ0 m1 +(aT )m2 )⊤ Σ0 (m1 −(aT )2 (K0 +a2 T 2 Σ0 )−1 Σ0 m1 +(aT )m2 )
×e
1
× e−2 T (−a
1
2
T (K0 +a2 T 2 Σ0 )−1 Σ0 m1 +am2 )⊤ Id (−a2 T (K0 +a2 T 2 Σ0 )−1 Σ0 m1 +am2 )
2
2
−1
⊤
2
2
−1
× e−2 (−(1−a)aT (K0 +a T Σ0 ) Σ0 m1 +(1−a)m2 ) Σε (−(1−a)aT (K0 +a T Σ0 ) Σ0 m1 +(1−a)m2 )
1 ⊤ Σ0 (K0 + a2 T 2 Σ0 )−1 K0
0d×d
⊤ 0d×1
m ,
= exp im
− m
0d×d
a2 T 2 Σ0 + K0
aT λ0
2
i.e. (φS , ITS ) is multivariate normally distributed.
30
4.3 Enlargement and Filtering
Hence, φS and ITS are independent. Now, let p ∈ Rd . Then by the independence of φS and ITS we
get
h ⊤
i
h ⊤ S
i
2 2
−1 S
E eip λ | ITS
= E eip (φ +λ0 +aT Σ0 (K0 +a T Σ0 ) (IT −aλ0 T )) | ITS
h ⊤ Si ⊤
2 2
−1 S
= E eip φ eip (λ0 +aT Σ0 (K0 +a T Σ0 ) (IT −aλ0 T )) ,
−1 S
−1 i.e. P(λ ∈ · | ITS ) = N λ0 + aT Σ0 K0 + a2 T 2 Σ0
(IT − aλ0 T ), K0 K0 + a2 T 2 Σ0
Σ0 a.s.
with
Q0 (· | iT ) = P(λ ∈ · | ITS ) on the set {ITS = iT }.
Theorem 4.3.15. Let µ0 (· | iT ) := Q0 (· | iT ). Then µn (· | iT , z1 . . . , zn ) follows a multivariate
normal distribution
µn (· | iT , z1 , . . . , zn )
2
−1
=N
(1 − a)
(Ln )
Kn−1 Σε
n
X
zk + Σ−1
0 λ0 + a(N
k=1
− n)hKn−1 iT
!
!
−1
, (Ln )
for n = 0, . . . , N and with
Ln := (1 − a)4 nh Σε K0−1 Kn−1 Σε + a2 T 2 K0−1 + Σ−1
0 .
Proof. We make use of the following abbreviations
L :=
−a2 (N − n)hKn−1 + Id = (1 − a)2 Kn−1 Σε ;
1
1
a2 Kn−1 + Id = Kn−1 Kn−1 ;
h
h
−1
2 2 2
Σ−1
0 (a N h Σ0 + K0 )K0 ;
iS
n
:=
aKn−1 iT ;
m
:=
aN hK0−1 iT + Σ−1
0 λ0 .
Dn
:=
An
:=
Therefore
QZ
n (· | θ, iT , yn )
Q0 (· | iT )
−1
S
2 −1
= N A−1
n+1 Dn+1 θ + in+1 − a Kn+1 yn , An+1 ;
= N L−1 m, L−1 .
We show by induction and using the definition of the Bayes operator that
µn (· | iT , hn )
(4.10)


!
!
!
−1
−1
n
n
n
X
X
X
−1 S
−1
.
=N
Dk A−1
D
+
L
D
y
−
D
A
i
+
m
,
D
A
D
+
L
n n
k
k k k
k k
k
k
k=1
k=1
k=1
31
4 Discrete-Time Consumption-Investment Problems
Initial step:
n=0:
n=1:
X since: µ0 (· | iT ) := Q0 (· | iT ) = N L−1 m, L−1
X since (note that y0 = 0):
(y0 = 0 and
0
P
k=1
· := 0);
µ1 (B | iT , z1 )
R Z
q0 (z1 | θ, iT , y0 )µ0 (dθ | iT )
B
= R Z
q0 (z1 | θ, iT , y0 )µ0 (dθ | iT )
Θ
=
R
B
R
Θ
=
R
√
⊤
−1
S ⊤
−1
S
1
det A1 det L − 21 (z1 −A−1
m) L(θ−L−1 m)
1 (D1 θ+i1 )) A1 (z1 −A1 (D1 θ+i1 ))− 2 (θ−L
dθ
e
(2π)d
√
−1
1
S
S ⊤
−1 m)⊤ L(θ−L−1 m)
det A1 det L − 21 (z1 −A−1
1 (D1 θ+i1 )) A1 (z1 −A1 (D1 θ+i1 ))− 2 (θ−L
e
dθ
d
(2π)
−1 S
−1
⊤
⊤
1
e− 2 (θ (D1 A1 D1 +L)θ−2θ (D1 (z1 −A1 i1 )+m)) dθ
B
R
−1 S
−1
1
⊤
⊤
e− 2 (θ (D1 A1 D1 +L)θ−2θ (D1 (z1 −A1 i1 )+m)) dθ
,
Θ
where in the last step we cancel all terms which do not depend on θ. In the next step we complete
the square, i.e.
−1 S
⊤
θ⊤ (D1 A−1
1 D1 + L)θ − 2θ (D1 (z1 − A1 i1 ) + m)
⊤ −1
S
S
= f1 (θ) − (D1 A−1
(D1 (z1 − A−1
D1 (z1 − A−1
1 D1 + L)
1 i1 ) + m)
1 i1 ) + m ,
where
⊤
−1 S
−1
f1 (θ) := θ−(D1 A−1
D
+L)
(D
(z
−A
i
)+m)
1
1
1
1
1 1
−1
S
−1
(D1 (z1 −A−1
× (D1 A−1
1 i1 )+m) .
1 D1 +L) θ−(D1 A1 D1 +L)
By canceling the correction term from completing the square, we finally obtain
p
µ1 (· | iT , z1 ) = q
i.e. µ1 (· | iT , z1 ) = N
|
det (D1 A−1
1 D1 +L)
d
(2π) 2
R
1
e− 2 f1 (θ) dθ
B
Z
det (D1 A−1
1 D1 + L)
d
(2π) 2
,
e
− 21 f1 (θ)
dθ
Θ
{z
=1
}
−1
−1 −1 S
−1
D1 A−1
D
+
L
(D
(z
−
A
i
)
+
m),
D
A
D
+
L
.
1
1
1
1
1
1
1 1
1
Inductive step: Let µn (· | iT , hn ) be given as in (4.10) (induction hypothesis).
32
4.3 Enlargement and Filtering
n → n + 1 : With the induction hypothesis it holds
µn+1 (B | iT , hn , zn+1 )
R Z
qn (zn+1 | θ, iT , yn )µn (dθ | iT , hn )
B
=R Z
qn (zn+1 | θ, iT , yn )µn (dθ | iT , hn )
Θ
=
− 12
R
e
B
− 12
R
e
Θ
n+1
P
⊤
θ
k=1
n+1
P
⊤
θ
θ−2θ
θ−2θ ⊤
Dk A−1
Dk+L
k
Dk A−1
Dk+L
k
k=1
⊤
n
−1
2
S
Dk A−1
iS
Dn+1 (zn+1−A−1
k +m
n+1 (in+1−a Kn+1 yn ))+Dn yn−
k
k=1
−1
S
2
Dn+1 (zn+1−A−1
n+1 (in+1−a Kn+1 yn ))+Dn yn−
P
n
P
Dk A−1
iS
+m
k
k
k=1
dθ
.
dθ
We complete again the square: By virtue of
−1
Dn+1 − Dn = a2 Dn+1 A−1
n+1 Kn+1
we can summarize the sums containing yn so that
!
n+1
X
−1
⊤
θ
Dk Ak Dk + L θ
(4.11)
k=1
− 2θ
⊤
S
2 −1
Dn+1 (zn+1 − A−1
n+1 (in+1 − a Kn+1 yn )) + Dn yn −

= fn+1 (θ) − 
n+1
X
!−1
Dk A−1
k Dk + L
k=1
Dn+1 (yn + zn+1 ) −
n
X
S
Dk A−1
k ik
k=1
n+1
X
k=1
!⊤
S

Dk A−1
k ik + m
× Dn+1 (yn + zn+1 ) −
where

fn+1 (θ) := θ −
×
n+1
X
Dk A−1
k Dk + L
k=1
n+1
X
× θ −
Dn+1 (yn + zn+1 ) −
!
n+1
X
S
Dk A−1
k ik
k=1
n+1
X
k=1
!
+m ,
!⊤
S

Dk A−1
k ik + m
Dk A−1
k Dk + L
k=1

!−1
+m
!
n+1
X
!−1
Dk A−1
k Dk + L
k=1
Dn+1 (yn + zn+1 ) −
n+1
X
k=1
!
S
.
Dk A−1
k ik + m
By canceling the terms which do not depend on θ we finally obtain
s n+1
P
det
Dk A−1
D
+L
k
k
R − 1 f (θ)
k=1
e 2 n+1 dθ
d
(2π) 2
B
µn+1 (B | iT , hn , zn+1 ) = s ,
n+1
P
−1
det
Dk Ak Dk + L Z
1
k=1
e− 2 fn+1 (θ) dθ
d
(2π) 2
|
Θ
{z
=1
}
33
4 Discrete-Time Consumption-Investment Problems
i.e.
µn+1 (· | iT , hn , zn+1 )

!−1
! n+1
!−1
n+1
n+1
X
X
X
S
.
=N 
Dk A−1
Dn+1(yn +zn+1 )−
Dk A−1
Dk A−1
k Dk +L
k ik +m ,
k Dk +L
k=1
k=1
k=1
With (4.11) we get
n
X
Dk A−1
k Dk
k=1
n
X
S
Dk A−1
k ik
=
(1 − a)4 nhΣε K0−1 Kn−1 Σε ;
=
(1 − a)2 anhΣε K0−1 Kn−1 iT ,
k=1
so that µn (· | iT , hn ) reduces to
−1
µn (· | iT , z1 , . . . , zn ) = N (Ln )−1 Dn yn + a(N − n)hKn−1 iT + Σ−1
.
0 λ0 , (Ln )
(II) Brownian Information
We consider Brownian information, i.e. IT = ITW .
Lemma 4.3.16. The initial distribution Q0 (· | iT ) in the filtering recursion (with Brownian
information) follows a multivariate normal distribution
Q0 (· | iT ) = Q0 (·) = N (λ0 , Σ0 ).
Proof. Observe that ITW and λ are independent. Hence,
1
E[exp(ip⊤ λ) | ITW ] = E[exp(ip⊤ λ)] = exp ip⊤ λ0 − p⊤ Σ0 p ,
2
i.e. P(λ ∈ · | ITW ) = N (λ0 , Σ0 ) with Q0 (· | iT ) = P(λ ∈ · | ITW ) on the set {ITW = iT }.
Theorem 4.3.17. Let µ0 (· | iT ) := Q0 (·). Then µn (· | iT , z1 , . . . , zn ) follows a multivariate normal
distribution
µn (· | iT , z1 , . . . , zn )
=N
−1
nhK0 Kn−1 + Σ−1
0
for n = 0, . . . , N − 1.
34
K0 Kn−1
n
X
k=1
−1
zk + Σ−1
0 λ0 − anhKn iT
!
,
−1
nhK0 Kn−1 + Σ−1
0
!
4.3 Enlargement and Filtering
Proof. We make use of the following abbreviations
An
:=
Dn
iW
n
:=
:=
1 −1
K Kn−1 ;
h n
K0 Kn−1 ;
aKn−1 iT .
Therefore
−1
−1
W
2 −1
,
A
QZ
D
θ
+
i
(·
|
θ,
i
,
y
)
=
N
A
−
a
K
y
n+1
n
n
T
n
n+1
n+1 .
n+1
n+1
As in the proof of Theorem 4.3.15 it can be shown by an induction that
µn (· | iT , hn )

!−1
!
!−1
n
n
n
X
X
X
−1
−1
−1
W
.
Dk A−1
=N 
Dn yn −
Dk A−1
Dk A−1
k Dk +Σ0
k ik +Σ0 λ0 ,
k Dk +Σ0
k=1
k=1
k=1
Then, the assertion follows by using
−1
Dn+1 − Dn = a2 Dn+1 A−1
n+1 Kn+1 .
(III) Drift Information
Let the investor have drift information, i.e. IT = ITλ .
Lemma 4.3.18. The initial distribution Q0 (· | iT ) in the filtering recursion (with drift information) follows a multivariate normal distribution
Q0 (· | iT )
−1 −1
2
2
2
λ
+ai
,(1
−
a)
Σ
a
Σ
+
(1
−
a)
Σ
=N Σ0 a2 Σ0 +(1 − a)2 Σε
(1 − a)2 Σε Σ−1
Σ0 .
0
ε
0
ε
T
0
Proof. We consider
φλ
:=
=
ITλ
=
−1 λ
(λ − λ0 ) − aΣ0 a2 Σ0 + (1 − a)2 Σε
(IT − aλ0 )
−1
−1
(1 − a)2 Σε a2 Σ0 + (1 − a)2 Σε
(λ − λ0 ) − a(1 − a)Σ0 a2 Σ0 + (1 − a)2 Σε
ε,
aλ + (1 − a)ε.
We show that φλ and ITλ are independent:
• φλ and ITλ are uncorrelated since
Cov(φλ , ITλ )
=a(1 − a)2Σε a2 Σ0 + (1 − a)2 Σε
=0d×d ;
−1
Cov(λ, λ) − a(1 − a)2 Σ0 a2 Σ0 +(1 − a)2 Σε
−1
Cov(ε, ε)
35
4 Discrete-Time Consumption-Investment Problems
⊤ ⊤
2d with m , m ∈ Rd . By the independence
• (φλ , ITλ ) is Gaussian: Let m = (m⊤
1
2
1 , m2 ) ∈ R
of λ and ε we obtain
λ φ
E exp im⊤ λ
I
2
2
2
−1
= exp(im⊤
λ0 )
1 (−(1 − a) Σε (a Σ0 + (1 − a) Σε )
× E[exp(i(((1 − a)2 (a2 Σ0 + (1 − a)2 Σε )−1 Σε m1 + am2 )⊤ λ))]
× E[exp(i(−a(1 − a)(a2 Σ0 + (1 − a)2 Σε )−1 Σ0 m1 + (1 − a)m2 )⊤ ε)]
1
(1 − a)2 Σε (a2 Σ0 + (1 − a)2 Σε )−1 Σ0
0d×d
0
m ,
= exp im⊤ d×1 − m⊤
0d×d
a2 Σ0 + (1 − a)2 Σε
aλ0
2
i.e. (φλ , ITλ ) is multivariate normally distributed.
Hence, φλ and ITλ are independent. Now, let p ∈ Rd . Then by the independence of φλ and ITλ we
get
E[exp(ip⊤ λ) | ITλ ]
= E[exp(ip⊤ (φλ + λ0 + aΣ0 (a2 Σ0 + (1 − a)2 Σε )−1 (ITλ − aλ0 ))) | ITλ ]
⊤
= E[exp(ip⊤ φλ )]eip
λ
(λ0 +aΣ0 (a2 Σ0 +(1−a)2 Σε )−1 (IT
−aλ0 ))
= exp(ip⊤ ((1 − a)2 Σε (a2 Σ0 + (1 − a)2 Σε )−1 λ0 + aΣ0 (a2 Σ0 + (1 − a)2 Σε )−1 ITλ ))
1
× exp(− p⊤ (1 − a)2 Σε (a2 Σ0 + (1 − a)2 Σε )−1 Σ0 p),
2
i.e.
P(λ ∈ · | ITλ )
−1
−1 λ
2
2
2
=N Σ0 a2 Σ0 +(1−a)2 Σε
((1−a)2 Σε Σ−1
Σ0
0 λ0 +aIT ), (1−a) Σε a Σ0 +(1−a) Σε
with Q0 (· | iT ) = P(λ ∈ · | ITλ ) on the set {ITλ = iT }.
Theorem 4.3.19. Let µ0 (· | iT ) := Q0 (· | iT ). Then µn (· | iT , z1 , . . . , zn ) follows a multivariate
normal distribution
µn (· | iT , z1 , . . . , zn )
=N
Σε Mn−1 Σ0 (1 − a)2
n
X
k=0
−1
zk + (1 − a)2 Σ−1
0 λ0 + aΣε iT
!
, (1 − a)2 Σε Mn−1 Σ0
!
for n = 0, . . . , N − 1 and with Mn := nh(1 − a)2 Σ0 Σε + a2 Σ0 + (1 − a)2 Σε .
Proof. The assertion follows by an induction using the filter recursion (in the same way as in
Theorem 4.3.15).
As seen in Theorem 4.3.15, Theorem 4.3.17 and Theorem 4.3.19, (µn ) depends on the past observations, z1 , . . . , zn , only through the number n and the sum of observations yn . We consequently
obtain the following result.
36
4.3 Enlargement and Filtering
Corollary 4.3.20. For n = 0, . . . , N − 1 the sum Yn =
n
P
Zk is a sufficient statistic for (µn ), i.e.
k=1
there exists a sequence of transition kernels (µ̂n ) from I × Z to Θ such that
µn (C | iT , hn ) = µ̂n (C | iT ,
n
X
zk )
k=1
for all C ∈ B(Θ), iT ∈ I and hn ∈ Z n .
Definition 4.3.21. For iT ∈ I and y ∈ Z we define
Z
Q̂Z
(B
|
i
,
y)
:=
QZ
T
n
n (B | θ, iT , y)µ̂n (dθ | iT , y)
for all B ∈ B(Z).
Note that the distributions QZ
n and µ̂n are of the following form
QZ
n (· | λ, IT , Yn )
µ̂n (· | IT , Yn )
= N (b + Aλ, D);
= N (c, E),
where the symmetric and non-singular matrices A, D, E ∈ Rd×d (D, E positive definite) and the
vectors b, c ∈ Rd are given in Table 4.1. The vectors b, c may depend on IT and Yn .
Theorem 4.3.22. For n = 0, . . . , N − 1 we have that Q̂Z
n (· | IT ,
n
P
Zk ) is a stochastic kernel
k=1
which determines the distribution of the observation Zn+1 given the information σ(IT ,
n
P
Zk ).
k=1
Moreover,
Q̂Z
n (· | IT ,
n
X
k=1
Zk ) = N (b + Ac, D + AEA),
where b, c, A, D and E are given in Table 4.1.
Proof. We prove the theorem in two steps. Firstly, we show in (i) that Q̂Z
n is a stochastic kernel
Z
with Q̂Z
n (· | IT , Yn ) = N (b + Ac, D + AEA). Then we conclude in (ii) that Q̂n is a conditional
n
P
Zk ).
distribution of Zn+1 given the information σ(IT ,
k=1
(i) Let B ∈ B(Z). Then
Q̂Z
n (B | IT , Yn )
Z Z
=
QZ
n (dzn+1 | θ, IT , Yn )µ̂n (dθ | IT , Yn )
B
=
Z Z
1
⊤
e− 2 (zn+1 −(b+Aθ))
B
D −1 (zn+1 −(b+Aθ))− 12 (θ−c)⊤ E −1 (θ−c)
(2π)d
√
det E det D
dzn+1 dθ.
We rearrange the exponent
(zn+1 − (b + Aθ))⊤ D−1 (zn+1 − (b + Aθ)) + (θ − c)⊤ E −1 (θ − c) = f θ (θ, zn+1 ) + f z (zn+1 ),
37
4 Discrete-Time Consumption-Investment Problems
(I) Stock price information
b
−1
hKn
(aITS − a2 Yn )
A
−1
(1 − a)2 hKn
Σε
D
−1
hKn
Kn+1
c
−1
−1 ((1 − a)2 K −1 Σ Y + a(N − n)hK −1 I S + Σ−1 λ )
((1 − a)4 nhΣε K0−1 Kn
Σε + (a2 T 2 K0−1 + Σ−1
ε n
0
n
n
0 ))
0
T
E
−1
−1
((1 − a)4 nhΣε K0−1 Kn
Σε + (a2 T 2 K0−1 + Σ−1
0 ))
(II) Brownian information
b
−1
hKn
(aITW − a2 Yn )
A
−1
hKn
K0
−1
hKn
Kn+1
D
−1 −1
−1
nhK0 Kn
+ Σ0
c
−1
−1 W
(K0 Kn
Yn − anhKn
IT + Σ−1
0 λ0 )
−1
nhK0 Kn
+ Σ−1
0
E
−1
(III) Drift information
b
0
A
hId
D
hId
c
−1 λ
Σε (nh(1 − a)2 Σ0 Σε + a2 Σ0 + (1 − a)2 Σε )−1 Σ0 ((1 − a)2 Yn + (1 − a)2 Σ−1
0 λ0 + aΣε IT )
E
(1 − a)2 Σε (nh(1 − a)2 Σ0 Σε + a2 Σ0 + (1 − a)2 Σε )−1 Σ0
Table 4.1: QZ
n (· | λ, IT , Yn ) = N (b + Aλ, D) and µ̂n (· | IT , Yn ) = N (c, E)
where
⊤
f θ (θ, zn+1 ) := θ − (AD−1 A + E −1 )−1 (AD−1 (zn+1 − b) + E −1 c) (AD−1 A + E −1 )
× θ − (AD−1 A + E −1 )−1 (AD−1 (zn+1 − b) + E −1 c) ;
f z (zn+1 ) := (zn+1 − (b + Ac))⊤ (D + AEA)−1 (zn+1 − (b + Ac)) .
By Fubini’s theorem we obtain
Q̂Z
n (B | IT , Yn )
Z Z
1 θ
1 z
1
√
e− 2 f (θ,zn+1 )− 2 f (zn+1 ) dzn+1 dθ
=
d
(2π) det E det D
B
Z p
det((AD−1 A + E −1 )−1 ) − 1 f z (zn+1 )
=
e 2
d p
(2π) 2 det(DE)
B
Z
1 θ
1
e− 2 f (θ,zn+1 ) dθ dzn+1 .
×
d p
−1
−1
−1
2
(2π)
det((AD A + E ) )
|
{z
}
=1
38
4.3 Enlargement and Filtering
Due to the positive definiteness of D and E the matrix D + AEA is also positive definite.
Since det(DE(AD−1 A + E −1 )) = det(AEA + D) it follows that
Q̂Z
n (· | IT , Yn ) = N (b + Ac, D + AEA) .
(4.12)
Moreover, Q̂Z
n (B | IT , Yn ) is obviously measurable w.r.t. σ(IT , Yn ) for each B ∈ B(Z).
(ii) Let B ∈ B(Z). Then
P(Zn+1 ∈ B | IT , Yn )
= E[E[1B (Zn+1 ) | λ, IT , Yn ] | IT , Yn ]


Z

= E  QZ
n (dzn+1 | λ, IT , Yn ) | IT , Yn
=
=
Z ZB
QZ
n (dzn+1 | θ, IT , Yn )µ̂n (dθ | IT , Yn )
B
Z
Q̂n (B
| IT , Yn ) a.s.,
i.e. Q̂Z
n (· | IT , Yn ) is a (version of the) conditional distribution of Zn+1 given IT , Yn with
Q̂Z
n (· | iT , yn ) = P(Zn+1 ∈ · | IT , Yn ) on the set {IT = iT , Yn = yn }.
From Theorem 4.3.22 together with Corollary 4.3.8 and Corollary 4.3.20 (sufficient statistic) we
obtain directly the following result.
Corollary 4.3.23. For n = 0, . . . , N − 1 we have
P(Zn+1 ∈ B | IT = iT , Z1 = z1 , . . . , Zn = zn ) = Q̂Z
n (B | iT ,
Thus, Q̂Z
n (· | IT ,
n
P
n
X
zk ).
k=1
Zk ) is a stochastic kernel which determines the distribution of the observation
k=1
Zn+1 given the history σ(IT , Z1 , . . . , Zn ).
4.3.3 Summary of the Enlargement and Filtering Results
In this subsection we briefly summarize the results obtained by the enlargement and filtering step
for all three types of insider information. From now on let the following assumption always be
satisfied.
Type of
insider information
IT = ITS
IT = ITW
IT = ITλ
d × d-Matrix
ΣIaT
−1 Σ
−a2 K0−1 + (1 − a)4 Σε (a2 (N h)2 K0 + K0 Σ−1
ε
0 K0 )
−a2 K0−1 + Σ0
(1 − a)2 Σε (a2 Σ0 + (1 − a)2 Σε )−1 Σ0
Table 4.2: "modified covariance matrix" ΣIaT (discrete-time)
39
4 Discrete-Time Consumption-Investment Problems
Assumption 4.3.24. The matrix (Id + N hΣIaT ) is positive definite, where ΣIaT is given in Table
4.2.
ΣIaT is symmetric and corresponds for a = 0 to the covariance matrix Σ0 of λ. For stock price and
Brownian information ΣIaT can for some parameterizations (i.e. for some choices of a ∈ [0, 1), T >
0, Σ0 , Σε ) be negative definite. Hence, ΣIaT is in general not a covariance matrix. We call ΣIaT a
modified covariance matrix.
Remark 4.3.25. Assumption 4.3.24 implies that the symmetric matrix (Id + nhΣIaT ) is positive
definite for all n ≤ N. Hence, (Id + nhΣIaT ) is non-singular. For instance, if the components of the
d-dimensional noise vector ε are independent, i.e. Σε = Id , then Assumption 4.3.24 is satisfied.
Type of
insider
information
d × d-matrix
C IT
d × d-matrix
CλIT
IT = ITS
a(Id + N hΣ0 )(a2 (N h)2 Σ0 + K0 )−1
IT = ITW
aK0−1
(1 − a)2 Σε (a2 (N h)2 Σ0 + K0 )−1
IT = ITλ
aΣ0 (a2 Σ0 + (1 − a)2 Σε )−1
Id
(1 − a)2 Σε (a2 Σ0 + (1 − a)2 Σε )−1
Table 4.3: Coefficients of iT and y in the expectation of Q̂Z
n (· | iT , y)
Theorem 4.3.26. Let Assumption 4.3.24 be satisfied and let the matrices C IT and CλIT be given
as in Table 4.3. Then for n = 0, . . . , N − 1 it holds that
Q̂Z
n (· | iT , y)
−1
−1
IT
IT
IT
IT
IT
IT
=N h Id + nhΣa
(Cλ λ0 + C iT + Σa y), h Id + nhΣa
(Id + (n + 1)hΣa ) .
Z
Furthermore, Q̂Z
n (· | iT , y) has a Lebesgue density q̂n (zn+1 | θ, iT , y), i.e.
Z
Z
Q̂n (B | iT , y) = q̂nZ (zn+1 | iT , y)dzn+1 , B ∈ B(Z).
B
The distribution Q̂Z
n (· | iT , y) follows a multivariate normal distribution. The expectation depends
explicitly and linearly on the insider information iT and the observation y. However, Q̂Z
n (· | iT , y)
does not depend on the history and on the unknown drift.
Proof. It is straightforward to show
h(CλIT λ0 + C IT iT + ΣIaT y) =
h(Id + (n + 1)hΣIaT )
=
(Id + nhΣIaT )(D + AEA);
(Id + nhΣIaT )(b + Ac),
where b, c, A, E are given in Table 4.1 (compare Theorem 4.3.22).
Remark 4.3.27. Contrary to exact drift and exact Brownian information (i.e. with convex factor
a = 1) Assumption 4.3.24 is never satisfied for exact stock price information. In the latter case
IT −1
(Id + N hΣIaT ) = 0d×d
the covariance matrix of Q̂Z
N −1 is singular, namely h(Id + (N − 1)hΣa )
(see also Remark 4.3.9 and Remark 4.4.10).
40
4.4 Reformulation as a Markov Decision Process
Remark 4.3.28. By the enlargement and filtering step we obtain the distribution
Q̂Z
n (· | iT ,
n
X
zk ).
k=1
In our approach the enlargement step is followed by the filtering step: Firstly, we include the
insider information (enlargement step) by determining
QZ
n (·
| θ, iT ,
d
X
zk )
k=1
and afterward we use the filter recursion (see Definition 4.3.11) in order to estimate the unknown
drift on the basis of the insider information and the observations (filtering step). However, these
two steps can also be interchanged:
In the first step the unknown drift is estimated only on the basis of the observations using the
following recursion
µ̃0 (C) ≔
µ̃n+1 (C | hn , zn+1 ) ≔
Q0 (C),
C ∈ B(Θ)
Φ̃n+1 (µ̃n (· | z1 , . . . , zn ), zn+1 )(C)
with Bayes operator
R
R
Φ̃n+1 (ρ, z)(C) ≔ C
Θ
q Z (· | θ)
q Z (z | θ)ρ(dθ)
q Z (z | θ)ρ(dθ)
.
Here
denotes the Lebesgue density of QZ (· | θ) (see (4.2)). In the second step the insider
information is incorporated by determining the conditional distribution of IT given λ = θ, Hn = hn .
We denote this distribution by QInT (· | θ, hn ) (see Lemma 4.6.4). With the conditional distribution
of IT given the history, namely
Z
QInT (· | hn ) := QInT (· | θ, hn )µ̃n (dθ | hn ),
the distribution Q̂Z
n (· | iT ,
n
P
zk ) can be derived using Bayes’ theorem.
k=1
S S S
If we need to make clear which type of insider information we consider, then we write Q̂SZ
n ,C ,Cλ ,Σa
Z
W
W
W
λZ
λ
λ
λ
for stock price information, Q̂W
n , C , Cλ , Σa for Brownian information and Q̂n , C , Cλ , Σa
I
I
I
Z
T
T
for drift information (instead of Q̂n , C T , Cλ , Σa ).
4.4 Reformulation as a Markov Decision Process
In this section we formulate an MDP by which we can solve the h-insider’s consumption-investment
problems.
As aforementioned, we denote by dom(U ) the domain of the utility function U, i.e. U : dom(U ) →
R. We consider either
dom(Uc ) = dom(Up ) = [0, ∞)
or
dom(Uc ) = dom(Up ) = (0, ∞) with lim Uc (x) = lim Up (x) = −∞.
xց0
xց0
41
4 Discrete-Time Consumption-Investment Problems
MDP
• State space E := dom(Up ) × I × Z, where dom(Up ) ⊆ R≥0 , I = Rd , Z = Rd , endowed
with the Borel σ-field B(E). An element of E, hereinafter also referred to as a state,
is denoted by (x, iT , y) ∈ E. An element x ∈ dom(Up ) is the current wealth, iT ∈ I
the (realized) value of the insider information and y ∈ Rd the current sum of the past
observations;
• Action space A := dom(Uc ) × Rd , dom(Uc ) ⊆ R≥0 , endowed with the Borel σ-field
B(A). An element of A, hereinafter also referred to as an action, is denoted by
(c, α) ∈ E, where c ∈ R≥0 is the rate at which one consumes (hc is the amount
of wealth consumed until the next trading time point) and αk , k = 1, . . . , d, with
α = (α1 , . . . , αd )⊤ ∈ Rd is the fraction of wealth which is invested into stock k;
• Possible state action combinations
(
Dn :=
(x, iT , y, c, α) : (x, iT , y) ∈ E, (c, α) ∈ A with
)
n
X
rh
⊤ −rh
P (x − hc)e (1 + α (e
R(Zn+1 ) − 1d )) ∈ dom(Up ) | IT = iT ,
Zk = y = 1 .
k=1
Dn is a subset of all state action combinations E × A. For a state (x, iT , y) ∈ E the set
of admissible actions is given by
Dn (x, iT , y) := {(c, α) ∈ A : (x, iT , y, c, α) ∈ Dn };
• Observation space Z = Rd endowed with the Borel σ-field B(Z). An element z ∈ Z denotes the observation of the h-investor which determines the price changes of the stocks;
• Transition function T : Dn × Z → E determines the next state via
T (x, iT , y, c, α, z) := (Tx (x, c, α, z), iT , y + z),
where Tx (x, c, α, z) := (x − hc)erh (1 + α⊤ (e−rh R(z) − 1d )) determines the new level
of wealth. The (realized) value of the insider information iT does not change over
time. The third component of the transition function updates the current sum of
observations y by the new observation z;
• Stochastic transition kernel Q̂Z
n from I × Z to Z, where
Q̂Z
n (· | iT , y) :=
N h(Id + nhΣIaT )−1 (CλIT λ0 + C IT iT + ΣIaT y), h(Id + nhΣIaT )−1 (Id + (n + 1)hΣIaT ) ,
is the distribution of the next observation given the (realized) value of the insider
information iT and the current sum of observations y. Q̂Z
n depends only through the
expectation linearly on iT and y; but is independent of x;
42
4.4 Reformulation as a Markov Decision Process
• One-step reward r : Dn → R is a measurable function which is given by the utility of
the consumption until the next trading time point, i.e.
r(x, iT , y, c, α) := hUc (c);
• Terminal reward g : E → R is a measurable function which is given by the utility of
the current wealth, i.e.
g(x, iT , y) := Up (x).
Remark 4.4.1. When we only consider the terminal wealth problem, we have to set r(x, iT , y, c, α)
≡ 0. The transition function which determines the next level of wealth is then given by
Tx (x, α, z) = xerh (1 + α⊤ (e−rh R(z) − 1d )).
For all iT , y ∈ Rd and n = 0, . . . , N − 1 the distribution Q̂Z
n (· | iT , y) follows a (regular) multivariate
normal distribution. Hence, we can only guarantee that the wealth is always in the domain of Up
by excluding short selling. Thus, we obtain the following result:
Lemma 4.4.2. For a given state (x, iT , y) ∈ E the set of admissible actions is independent of
iT , y and n
(
{(c, α) ∈ A : x − ch ∈ dom(Up ) and α ∈ D} if x > 0;
Dn (x, iT , y) = D(x) :=
A ∩ {(0, 0)}
otherwise
with D := {α ∈ [0, 1]d : α⊤ 1d ≤ 1} (Recall that we set α = 0d×1 for x − hc = 0, see (4.5)).
The transition function depends on the chosen and admissible action (c, α). Therefore, we are
able to control the MDP. In the theory of MDPs admissible actions are given by decision rules,
where a decision rule at time tn = nh is a measurable mapping fn : E → A such that fn (x, iT , y) ∈
Dn (x, iT , y) for all (x, iT , y) ∈ E. We denote by Fn the set of all decision rules at time tn .
Furthermore, we call a sequence of decision rules π := (f0 , . . . , fN −1 ) an N -stage Markovian
policy. For a Markovian policy π and a state (x, iT , y) ∈ E we define the associated expected
reward by
#
"N −1
X
r(Xk , IT , Yk , fk (Xk , IT , Yk )) + g(XN , IT , YN ) .
Vnπ (x, iT , y) := Eπn(x,iT ,y)
k=n
Eπn(x,i ,y) [·]
T
Here
denotes the expectation under the Markovian policy π when starting at time
tn = nh in state (x, iT , y) ∈ E and is induced by the transition probabilities of the MDP. Thus,
Vnπ (x, iT , y) is the expected reward over the remaining time from tn = nh to tN = T = N h when
starting at time tn in state (x, iT , y) and using policy π. Moreover, for a function v ∈ M(E) :=
{v : E → [−∞, ∞) : v measurable} we introduce the reward operator Tn,(c,α) and the maximal
reward operator Tn corresponding to the MDP
Z
Tn,(c,α) v(x, iT , y) := hUc (c) + v(Tx (x, c, α, z), iT , y + z)Q̂Z
n (dz | iT , y),
Tn v(x, iT , y)
:=
sup
(c,α)∈D(x)
Tn,(c,α) v(x, iT , y).
43
4 Discrete-Time Consumption-Investment Problems
The following lemma states that the reward iteration holds.
Lemma 4.4.3. Let π := (f0 , . . . , fN −1 ) be an N -stage Markovian policy and (x, iT , y) ∈ E. Then
for n = 0, 1, . . . , N − 1 it holds:
VN π (x, iT , y) = Up (x)
and
Vnπ (x, iT , y) = Tn,fn V(n+1)π (x, iT , y).
Proof. The assertion follows analogously to Theorem 2.3.4 in Bäuerle & Rieder [2011] by using
P(Zn+1 ∈ B | IT , Z1 , . . . , Zn ) = P(Zn+1 ∈ B | IT ,
n
X
k=1
Zk ) = Q̂Z
n (B | IT ,
n
X
k=1
Zk ), B ∈ B(Rd ). (4.13)
For n = 0, . . . , N − 1 we define the value function of the MDP. Let (x, iT , y) ∈ E. Then
Vn (x, iT , y) :=
sup
Vnπ (x, iT , y),
π∈ΠN −n
where ΠN −n denotes the set of all (N − n)-stage Markovian policies. Vn is the maximal expected reward (under Markovian policies) over the remaining time from tn to T when starting
in (x, iT , y). A policy π ∗ ∈ ΠN is called optimal for the N -stage MDP if
V0π∗ (x, iT , y) = V0 (x, iT , y),
for all (x, iT , y) ∈ E.
An admissible consumption-investment strategy π = (cn , αn )n∈{0,...,N −1} is in general not Markovian. However, by virtue of to the definition of the set Ah0 the strategy π is admissible for the MDP
(i.e. (cn , αn ) ∈ D(x)). On the other hand, it follows from the wealth recursion that for a fixed
initial wealth x0 the wealth xn , obtained under a Markovian policy π = (f0 , . . . , fN −1 ), can be
written as a function of iT , z1 , . . . , zn . The consumption-investment strategy (ĉn , α̂n )n∈{0,...,N −1}
with
!
n
X
zk
(ĉn (iT , z1 , . . . , zn ), α̂n (iT , z1 , . . . , zn )) := fn xn , iT ,
k=1
is obviously FIT Z -adapted and an admissible strategy for the h-insider. Hence, a Markovian policy π = (f0 , . . . , fN −1 ) is always an admissible strategy for the consumption-investment problem.
For more details on MDPs we refer to Bäuerle & Rieder [2011] and Hinderer [1970].
In Theorem 4.4.4 we show that the value V0 of the MDP with initial state (x0 , iT , 0) coincides with the value V0∗ of the consumption-investment problem and moreover an optimal
Markovian policy defines an optimal consumption-investment strategy. Hence, we are able to
solve the consumption-investment problem by means of the above MDP.
Theorem 4.4.4.
a) Let π = (cn , αn )n∈{0,...,N −1} be an admissible consumption-investment
strategy. Then it holds:
∗
V0π
(x0 , iT ) =
Z
...
Z X
N
k=1
π
Z
hUc (ck−1 ) + Up (XN
)Q̂Z
N −1 (dzN | iT , yN −1 ) . . . Q̂0 (dz1 | iT , 0).
Moreover, let π = (cn , αn )n∈{0,...,N −1} be a Markovian policy. Then it holds:
∗
V0π
(x0 , iT )
44
= V0π (x0 , iT , 0).
4.4 Reformulation as a Markov Decision Process
b) For the initial state (x0 , iT , 0) it holds:
V0∗ (x0 , iT ) = V0 (x0 , iT , 0).
Proof.
a) Firstly, note that the admissible consumption-investment strategy π is FIT Z -adapted
so that we can write (cn , αn ) and the wealth Xnπ (see wealth recursion) as functions of
IT , Z1 , . . . , Zn . Moreover, recall that the distribution of Xnπ depends through Z1 , . . . , Zn on
the unknown drift (compare Remark 4.2.5). By using the tower property of the conditional
expectation, namely
E
"
N
X
k=1
π
hUc (ck−1 ) + Up (XN
) | IT
" " "
=E E E . . . E
"
N
X
k=1
#
#
#
#
#
π
hUc (ck−1 ) + Up (XN
) | λ, IT , Z1 , . . . , ZN −1 . . . | λ, IT , Z1 | λ, IT | IT ,
we get
∗
V0π
(x0 , iT ) :=
=
Z
E
"
"
N
X
k=1
...
π
hUc (ck−1 ) + Up (XN
)|
Z X
N
k=1
X0π
#
= x0 , λ = θ, IT = iT Q0 (dθ | iT )
Z
π
hUc (ck−1 ) + Up (XN
)QINT−1
(dzN | θ, iT , z1 , . . . , zN −1 ) . . .
. . . QI0T Z (dz1 | θ, iT )Q0 (dθ | iT ).
In order to simplify the notation we define for n = 1, . . . , N − 1
vn (iT , z1 , . . . , zn , θ) :=
Z X
"
N
Z
π
(dzN | θ, iT , z1 , . . . , zN −1 ) . . .
...
hUc (ck−1 ) + Up (XN
)QINT−1
k=1
IT Z
. . . Qn+1 (dzn+2
With yn =
n
P
| θ, iT , z1 , . . . , zn+1 )QInT Z (dzn+1 | θ, iT , z1 , . . . , zn ).
zk the assertion follows by applying N -times Lemma 4.3.12
k=1
∗
(x0 , iT )
V0π
"
=
v1 (iT , z1 , θ)QI0T Z (dz1 | θ, iT )µ0 (dθ | iT )
$
=
v1 (iT , z1 , θ)µ1 (dθ | iT , z1 )QI0T Z (dz1 | θ′ , iT )µ0 (dθ′ | iT )
"
=
v1 (iT , z1 , θ)µ1 (dθ | iT , z1 )Q̂Z
0 (dz1 | iT , 0)
$
=
v2 (iT , z1 , z2 , θ)QI1T Z (dz2 | θ, iT , z1 )µ1 (dθ | iT , z1 )Q̂Z
0 (dz1 | iT , 0)
...
45
4 Discrete-Time Consumption-Investment Problems
...
=
Z
=
Z
=
=
...
$
Z
µN −2 (dθ | iT , z1 , . . . , zN −2 )Q̂Z
N −3 (dzN −2 | iT , yN −3 ) . . . Q̂0 (dz1 | iT , 0)
&
...
vN −1 (iT , z1 , . . . , zN −1 , θ)µN −1 (dθ | iT , z1 , . . . , zN −1 )
Z
(dzN −1 | θ′ , iT , z1 , . . . , zN −2 )µN −2 (dθ′ | iT , z1 , . . . , zN −2 )
QINT−2
Z
...
Z
...
|
Z
(dzN −1 | θ, iT , z1 , . . . , zN −2 )
vN −1 (iT , z1 , . . . , zN −1 , θ)QINT−2
$
Z
Q̂Z
N −3 (dzN −2 | iT , yN −3 ) . . . Q̂0 (dz1 | iT , 0)
vN −1 (iT , z1 , . . . , zN −1 , θ)µN −1 (dθ | iT , z1 , . . . , zN −1 )
Z X
N
k=1
Z
Z
Q̂Z
N −2 (dzN −1 | i, yN −2 )Q̂N −3 (dzN −2 | iT , yN −3 ) . . . Q̂0 (dz1 | iT , 0)
π
Z
hUc (ck−1 ) + Up (XN
)Q̂Z
N −1 (dzN | iT , yN −1 ) . . . Q̂0 (dz1 | iT , 0) .
{z
}
=:V̄0π (x0 ,iT ,0)
Let π be a Markovian policy. In particular, π is also an admissible consumption-investment
strategy. It then follows directly from the definition of Vnπ and the reward iteration in
Lemma 4.4.3 that
V0π (x0 , iT , 0) = V̄0π (x0 , iT , 0).
∗ (x , i ) = V̄ (x , i , 0), the assertion follows.
Since by the first part of the proof we have V0π
0 T
0π 0 T
b) Let π = (cn , αn )n∈{0,...,N −1} ∈ Ah0 . Since an admissible consumption-investment strategy
is FIT Y -adapted, there exists a measurable function gn : Rd(n+1) → R≥0 × D such that
(cn , αn ) = gn (IT , Z1 , . . . , Zn ), n = 0, . . . , N − 1, i.e. π might be history dependent. By the
first part of a) we get
∗
sup V0π
(x0 , iT ) = sup V̄0π (x0 , iT , 0).
π∈Ah
0
π∈Ah
0
Due to the Markovian structure of the state process (Xn , IT , Yn )n∈{0,...,N } it follows by
Theorem 2.2.3 in Bäuerle & Rieder [2011] that the maximal expected reward cannot be
improved by using history dependent policies, i.e.
sup V̄0π (x0 , iT , 0) = sup V̄0π (x0 , iT , 0).
π∈ΠN
π∈Ah
0
By the second part of a) we finally obtain
sup V̄0π (x0 , iT , 0) = sup V0π (x0 , iT , 0).
π∈ΠN
π∈ΠN
Hence
V0∗ (x0 , iT ) := sup V0π (x0 , iT ) = sup V̄0π (x0 , iT , 0) = sup V0π (x0 , iT , 0) =: V0 (x0 , iT , 0).
π∈Ah
0
π∈Ah
0
π∈ΠN
46
4.4 Reformulation as a Markov Decision Process
The following Lemma is used in order to show that the value function is well-defined:
Lemma 4.4.5. There exist constants b1 , . . . , b4 ∈ R≥0 such that it holds:
(i) hUc (c) ≤ b1 (1 + xeb1 |iT |+b1 |y| ) for all (x, iT , y, c, α) ∈ Dn ;
(ii) Up (x) ≤ b2 (1 + xeb2 |iT |+b2 |y| ) for all (x, iT , y) ∈ E;
(iii) For all n ∈ {0, . . . , N − 1} and (x, iT , y, c, α) ∈ Dn
Z
b3 1 + (x − ch)erh 1 + α⊤ e−rh R(z) − 1d eb3 |iT |+b3 |y+z| Q̂Z
n (dz | iT , y)
≤ b4 1 + xeb4 |iT |+b4 |y| ,
where |x|, x ∈ Rd , denotes the sum of the componentwise absolute value, i.e. |x| :=
d
P
k=1
|xk |.
Proof. (i) and (ii) are clear since any concave function can be bounded from above by an affine
function. For the sake of notation simplicity we show (iii) only for the case d = 1. The distribution
Q̂Z
n is of the form
Q̂Z
n (· | iT , y) = N (c1 iT + c2 y + c3 , c4 )
for some c1 , . . . , c3 ∈ R, c4 ∈ R>0 which may depend on n ∈ {0, . . . , N − 1} (compare Theorem
4.3.26). Now let b3 ∈ R≥0 . Then
Z
b3 (1 + (x − ch)erh (1 + α(e−rh R(z) − 1))eb3 |iT |+b3 |y+z| )Q̂Z
n (dz | iT , y)
Z
σ2
≤b3 + b3 xerh (1 + eσz−(r+ 2 )h )eb3 |iT |+b3 |y+z| Q̂Z
n (dz | iT , y)
Z
σ2
≤b3 + b3 xerh+b3 |iT | (1 + eσz−(r+ 2 )h )(eb3 (y+z) + e−b3 (y+z) )Q̂Z
n (dz | iT , y)
2
1
σ2
=b3 + b3 xerh+b3 |iT | e(σc2 +b3 c2 +b3 )y+(σ+b3 )c1 iT +(σ+b3 )c3 + 2 (σ+b3 ) c4 −(r+ 2 )h
1
+ e(σc2 −b3 c2 −b3 )y+(σ−b3 )c1 iT +(σ−b3 )c3 + 2 (σ−b3 )
+e
(b3 c2 +b3 )y+b3 c1 iT +b3 c3 + 21 b23 c4
+e
2
2
c4 −(r+ σ2 )h
−(b3 c2 +b3 )y−b3 c1 iT −b3 c3 + 12 b23 c4
≤b3 + 4b3 xerh+b3 |iT | eb̃(|iT |+|y|+1) ,
where we choose b̃ as the maximum of the absolute values of the constants which appear. Finally,
we obtain the statement by choosing b4 appropriately (since n ∈ {0, . . . , N − 1} we can choose b4
independent of n). The multidimensional case with iT , y, z ∈ Rd can be proven in exactly the
same way.
Theorem 4.4.6 states that the Bellman equation holds and the value V0∗ of the consumptioninvestment problem can be determined recursively.
Theorem 4.4.6. For the consumption-investment problem it holds:
a) The value function Vn (x, iT , y) is bounded from above by an affine function, strictly increasing, strictly concave and continuous in x ∈ dom(Up ) for all n ∈ {0, . . . , N }, iT ∈ I and
y ∈ Z.
47
4 Discrete-Time Consumption-Investment Problems
b) For n = 0, . . . , N − 1 the value function can be computed recursively by the Bellman equation.
Let (x, iT , y) ∈ E. Then
VN (x, iT , y) = Up (x)
Vn (x, iT , y) =
sup
(c,α)∈D(x)
Z
Z
hUc (c) + Vn+1 (Tx (x, c, α, z), iT , y + z)Q̂n (dz | iT , y) .
c) For n = 0, . . . , N − 1 there exist maximizers fn∗ of Vn+1 (i.e. Tn Vn+1 = Tn,fn∗ Vn+1 ) and
∗
h
(f0∗ , f1∗ , . . . , fN
−1 ) is an optimal strategy for the consumption-investment problem (P ).
Proof. We apply Theorem 2.3.8 (structure theorem) in Bäuerle & Rieder [2011]. Therefore, we
show that the structure assumptions (SAN )
(i) Up ∈ M;
(ii) For all v ∈ M there exists a maximizer fn of v (i.e. Tn v = Tn,fn v) with fn ∈ ∆n ;
(iii) If v ∈ M, then Tn v is well-defined and Tn v ∈ M
are satisfied with
(
M :=
v ∈ M(E) : v(x, iT , y) ≤ b(1 + xeb|iT |+b|y| ) for some b ∈ R≥0 ,
x 7−→ v(x, iT , y) is strictly increasing, strictly concave and continuous
)
for all (iT , y) ∈ I × Z, and v(0, iT , y) := −∞ if 0 < dom(Up )
and ∆n = Fn . First note that it follows directly from Lemma 4.4.5 that Tn v(x, iT , y) < ∞ for all
(x, iT , y) ∈ E and
Tn v(x, iT , y) ≤ b 1 + xeb(|iT |+|y|)
for some b ∈ R≥0 . We consider Tn v and plug in the transition function
Tn v(x, iT , y)
Z
Z
hUc (c) + v(Tx (x, c, α, z), iT , y + z)Q̂n (dz | iT , y)
=
sup
(c,α)∈D(x)
=
sup
(c,α)∈D(x)
Z rh
⊤ −rh
Z
hUc (c) + v (x − ch)e (1 + α (e
R(z) − 1d )), iT , y + z Q̂n (dz | iT , y) .
We use the transformation φ := (x − ch)α and adapt the set of admissible actions according to
this transformation
D̃(x) = {(c, φ) ∈ dom(Uc ) × Rd : 0 ≤ ch ≤ x and 0 ≤ φk ≤ x − ch, k = 1, . . . , d, φ⊤ 1d ≤ x − ch}.
Firstly, we show that
Z (x, c, φ) 7−→ v erh ((x − ch) + φ⊤ (e−rh R(z) − 1d )), iT , y + z Q̂Z
n (dz | iT , y)
|
{z
}
=:u(x,c,φ)
48
4.4 Reformulation as a Markov Decision Process
is strictly concave over the (adapted) set of possible state action combinations
D̃n := {(x, iT , y, c, φ) : (x, iT , y) ∈ E, (c, φ) ∈ dom(Uc ) × Rd with (c, φ) ∈ D̃(x)}
(for fixed (iT , y)). For this purpose, let η ∈ (0, 1) and (x, iT , y, c, φ), (x′ , iT , y, c′ , φ′ ) ∈ D̃n with
(x, iT , y, c, φ) , (x′ , iT , y, c′ , φ′ )
(there exists at least one entry in which the vectors are not equal). By the convexity of D̃n we
get
η(x, iT , y, c, φ) + (1 − η)(x′ , iT , y, c′ , φ′ ) ∈ D̃n ,
(4.14)
i.e. η(c, φ) + (1 − η)(c′ , φ′ ) ∈ D̃(ηx + (1 − η)x′ ). Since
ηx + (1 − η)x′ − (ηc + (1 − η)c′ )h + (ηφ + (1 − η)φ′ )⊤ (e−rh R(z) − 1d )
= η(x − ch + φ⊤ (e−rh R(z) − 1d )) + (1 − η)(x′ − c′ h + (φ′ )⊤ (e−rh R(z) − 1d ))
we obtain by the concavity of x 7−→ v(x, iT , y) that
u(ηx + (1 − η)x′ , ηc + (1 − η)c′ , ηφ + (1 − η)φ′ ) ≥ ηu(x, c, φ) + (1 − η)u(x′ , c′ , φ′ ).
Hence, u is concave. Moreover, since Q̂Z
n follows a multivariate normal distribution we have
Z
1{x−ch+φ⊤ (e−rh R(z)−1d )=x′ −c′ h+(φ′ )⊤ (e−rh R(z)−1d )} Q̂Z
n (dz | iT , y) = 0
(Q̂Z
n -zero set). Therefore, the strict concavity of v in the first component finally implies the strict
concavity of
Z (x, c, φ) 7−→ v erh (x − ch + φ⊤ (e−rh R(z) − 1d )), iT , y + z Q̂Z
(4.15)
n (dz | iT , y)
over D̃n for all fixed (iT , y). Now we show that M and ∆n satisfy (SAN ):
(i) Up ∈ M holds due to the definition of a utility function.
(ii) Let v ∈ M and (x, iT , y) ∈ E. In order to split the maximization over c and φ we make use
of the iterated supremum, i.e.
Tn v(x, iT , y)
=
sup
(c,φ)∈D̃(x)
= sup
hUc (c) +
sup
0≤ch≤x φ∈A(c,x)
Z
rh
v e
hUc (c) +
Z
⊤
−rh
((x − ch) + φ (e
rh
v e
⊤
R(z) − 1d )), iT , y + z
−rh
((x − ch) + φ (e
Q̂Z
n (dz
R(z) − 1d )), iT , y + z
| iT , y)
Q̂Z
n (dz
| iT , y)
with A(c, x) := {φ ∈ Rd : 0 ≤ φk ≤ x − ch, k = 1, . . . , d, φ⊤ 1d ≤ x − ch}. We consider the two
maximization problems within Tn v separately:
(I) Maximization over φ ∈ A(c, x):
Due to (4.15) the mapping
Z φ 7−→ v erh (x − ch + φ⊤ (e−rh R(z) − 1d )), iT , y + z Q̂Z
n (dz | iT , y)
is strictly concave over A(c, x) for fixed c, x. Moreover, A(c, x) is compact. Hence, there
always exists a unique maximizer φn which depends on the state (x, iT , y).
49
4 Discrete-Time Consumption-Investment Problems
(II) Maximization over c ∈ 0, hx :
For a fixed x ∈ dom(Up ) let φ ∈ A(c, x), φ′ ∈ A(c′ , x), c, c′ ∈ 0, hx with c , c′ and η ∈ (0, 1).
Then ηφ + (1 − η)φ′ ∈ A(ηc + (1 − η)c′ , x) and consequently
Z v erh (x−(ηc+(1−η)c′ )h+φ⊤ (e−rh R(z)−1d )), iT , y+z Q̂Z
sup
n (dz | iT , y)
φ∈A(ηc+(1−η)c′,x)
≥
Z
Z
v erh (x − (ηc + (1−η)c′ )h + (ηφ + (1 − η)φ′ )⊤ (e−rh R(z) − 1d )), iT , y+z Q̂Z
n (dz | iT , y)
v erh (x − ch + φ⊤ (e−rh R(z) − 1d )), iT , y + z Q̂Z
n (dz | iT , y)
Z + (1 − η) v erh (x − c′ h + (φ′ )⊤ (e−rh R(z) − 1d )), iT , y + z Q̂Z
n (dz | iT , y),
>η
(4.16)
where we use (4.15) (strict concavity) in order to get the last inequality. Since this holds for
all φ ∈ A(c, x), φ′ ∈ A(c′ , x) and the maxima over φ ∈ A(c, x), φ′ ∈ A(c′ , x) exist we obtain the
strict concavity of
Z v erh (x − ch + φ⊤ (e−rh R(z) − 1d )), iT , y + z Q̂Z
c 7−→ Uc (c)h + sup
n (dz | iT , y)
φ∈A(c,x)
over 0, hx , i.e. we maximize a strictly concave function over a compact set. Thus, for
each state (x, iT , y) there exists a unique maximizer cn . If 0 < dom(Uc ) = dom(Up ) (with
Uc (0)
= Up (0) = −∞) it is obvious that the maximum cannot be attained at the boundary
of 0, hx , i.e.
Z Z
rh
⊤ −rh
hUc (c)+ v e ((x−ch)+φ (e
R(z)−1d )), iT , y+z Q̂n (dz | iT , y)
sup
sup
0≤ch≤x φ∈A(c,x)
= sup
sup
0<ch<x φ∈A(c,x)
Z (dz
|
i
,
y)
.
hUc (c)+ v erh ((x−ch)+φ⊤ (e−rh R(z)−1d )), iT , y+z Q̂Z
T
n
Hence, it follows from the maximization problems (I) and (II) that for v ∈ M there exists a
maximizer fn = (cn , αn ) with αn = (x−cn h)φn (αn := 0 for x−cn h = 0) so that Tn v = Tn,fn v.
The measurability of fn follows from a measurable selection theorem (see Bäuerle & Rieder
[2011] Theorem A.2.4), i.e. fn ∈ ∆n .
(iii) Recall the strict concavity in (4.15) and the convexity of the set D̃n (see (4.14)). Then the
strict concavity of x 7−→ Tn v(x, iT , y) follows using similar arguments as in (4.16). Furthermore, for x > 0 the transformation c̃ = xc is well-defined so that
Tn v(x, iT , y)
= sup Uc (c̃x)h+ sup
1
0≤c̃≤ h
α∈D
Z
(4.17)
v x(1−c̃h)erh (1+α⊤ (e−rh R(z)−1d )), iT , y+z Q̂Z
n (dz | iT , y).
Now we see that x 7−→ Tn v(x, iT , y) is strictly increasing due to the monotonicity of v in the
first component (note that 1 + α⊤ (e−rh R(z) − 1d ) > 0) and by virtue of the existence of the
maximizers. Finally, the concavity of x 7−→ Tn v(x, iT , y) implies its continuity over (0, ∞).
Thus, it is left to show the right continuity in x = 0 (if dom(Up ) = [0, ∞)). For x = 0 the
only admissible action is given by (c, α) = (0, 0). Hence,
Z
Tn v(0, iT , y) = Uc (0)h + v(0, iT , y + z)Q̂Z
n (dz | iT , y).
50
4.4 Reformulation as a Markov Decision Process
We select a sequence (xm )m∈N > 0 converging to zero from above with xm < b̃ for some
b̃ ∈ R>0 . For every xm there exists a maximizer (c̃m , αm ) ∈ [0, h1 ] × D of (4.17) (note that
c̃m , αm are bounded). Then we consider
lim Tn v(xm , iT , y)
m→∞
= lim
m→∞
Uc (c̃m xm )h
+
Z
!
rh
⊤ −rh
Z
v xm (1−c̃m h)e (1+αm (e
R(z)−1d )),iT ,y+z Q̂n (dz | iT ,y) .
Since x 7−→ v(x, iT , y) and x 7→ Uc (x) are continuous the continuity of Tn v(x, iT , y) in x = 0
follows when we are able to apply dominated convergence in order to interchange the limit
and the integral. However, this can be done since due to the boundedness of c̃m , αm , xm and
by virtue of Lemma 4.4.5 there exists a Q̂Z
n -integrable upper bound (independent of m) of
⊤ −rh
v xm (1 − c̃m h)erh (1 + αm
(e
R(z) − 1d )), iT , y + z .
Eventually, due to the existence of the maximizer fn we have Tn v(x, iT , y) ∈ M(E) (compare
Bäuerle & Rieder [2011] Remark 2.3.2 or Proposition 2.4.11). All in all, Tn v ∈ M.
Therefore, the assumptions (i) to (iii) of (SAN ) are satisfied with M and ∆n . Then the statements
a) to c) follow from the structure theorem (see Bäuerle & Rieder [2011]).
Arbitrage Opportunity
Definition 4.4.7. A self-financing and FIT Y -adapted strategy π = (cn , αn )n∈{0,...,N −1} is called
an arbitrage opportunity if the corresponding wealth process satisfies: X0π = 0,
π
P(XN
≥ 0) = 1
and
π
P(XN
> 0) > 0.
A financial market is said to be free of arbitrage if there does not exist an arbitrage opportunity.
In Theorem 3.1.5 Bäuerle & Rieder [2011] show that the market is free of arbitrage if and only if
there is locally no arbitrage opportunity:
Lemma 4.4.8. The following statements are equivalent:
a) The (discrete-time) financial market is free of arbitrage.
b) For n = 0, . . . , N − 1 and for all FInT Y -measurable φn ∈ Rd it holds
−rh
⊤ −rh
P φ⊤
(e
R(Z
)
−
1
)
≥
0
=
1
⇒
P
φ
(e
R(Z
)
−
1
)
=
0
= 1,
n+1
n+1
d
d
n
n
where φn denotes the amount of wealth invested into the stock.
By Lemma 4.4.8 we can show that the introduced discrete-time financial market is free of arbitrage.
Theorem 4.4.9. The (discrete-time) financial market is free of arbitrage.
51
4 Discrete-Time Consumption-Investment Problems
Proof. Let w.l.o.g. c ≡ 0, i.e. we consider the terminal wealth problem without consumption. Suppose there exists an arbitrage opportunity. From Lemma 4.4.8 and property (4.13)
it follows that there exists an FIT Z -adapted strategy φ∗ = (φ∗n )n∈{0,...,N −1} such that for some
n ∈ {0, 1, . . . , N − 1} and an (iT , z1 , . . . , zn ) ∈ I × Z n it holds
Z
1{φ∗n (iT ,z1 ,...,zn )⊤ (e−rh R(z)−1d )≥0} Q̂Z
n (dz | iT , y) = 1 and
Z
1{φ∗n (iT ,z1 ,...,zn )⊤ (e−rh R(z)−1d )>0} Q̂Z
(4.18)
n (dz | iT , y) > 0.
with y =
n
P
zk . By Theorem 4.4.6 there exists for the state (x, iT , y) a unique maximizer of
k=1
Z
Vn+1 xerh (1 + α⊤ (e−rh R(z) − 1d )), iT , y + z Q̂Z
n (dz | iT , y)
α∈D
Z
=
sup
Vn+1 erh (x + φ⊤ (e−rh R(z) − 1d )), iT , y + z Q̂Z
n (dz | iT , y)
Tn Vn+1 (x, iT , y) =
sup
φ∈D̃(x)
with
D̃(x) := {φ ∈ Rd : 0 ≤ φk ≤ x, k = 1, . . . , d, φ⊤ 1d ≤ x}
Z
d
={φ ∈ R : 1{x+φ⊤ (e−rh R(z)−1d )≥0} Q̂Z
n (dz | iT , y) = 1}.
We denote the maximizer over D̃(x) by fn∗ (x, iT , y). Then
Z
1{x+(fn∗ (x,iT ,y)+φ∗n (iT ,z1 ,...,zn ))⊤ (e−rh R(z)−1d )≥0} Q̂Z
n (dz | iT , y) = 1,
i.e. fn∗ + φ∗n ∈ D̃(x). By Theorem 4.4.6 the function Vn+1 (·, iT , y) is strictly increasing. Hence,
by (4.18) we obtain
Z
Vn+1 erh (x + fn∗ ⊤ (e−rh R(z) − 1d )), iT , y + z Q̂Z
n (dz | iT , y)
Z
<
Vn+1 erh (x + ((fn∗ + φ∗n )⊤ (e−rh R(z) − 1d )), iT , y + z Q̂Z
n (dz | iT , y).
This is a contradiction to fn∗ being the unique maximizer and the fact that we cannot improve
the performance by using history dependent policies. Hence, the discrete-time financial market
is free of arbitrage.
Remark 4.4.10. For an investor with exact stock price information the covariance matrix of
Q̂Z
N −1 is zero. This means that the last decision at time t = (N − 1)h is deterministic even when
the investor does not know the drift. In this case the set of admissible actions DN −1 (x, iT , y)
depends also on (iT , y), is not compact and there does not exist a maximizer. The investor has
then an arbitrage opportunity by short selling a stock and the bond, respectively.
4.5 Solutions for Logarithmic and Power Utility Functions
In this section we solve the consumption-investment problems for an h-insider with either logarithmic or power utility functions.
52
4.5 Solutions for Logarithmic and Power Utility Functions
4.5.1 Logarithmic Utility Functions
We consider an investor with logarithmic utility functions, i.e.
Uc (x) = Up (x) = log(x)
with dom(Uc ) = dom(Up ) = (0, ∞).
Theorem 4.5.1. For the consumption-investment problem it holds:
a) For n = 0, . . . , N the h-insider’s value function can be computed recursively
Vn (x, iT , y) = (1 + (N − n)h) log(x) + dn (iT , y),
(x, iT , y) ∈ E,
(4.19)
where the sequence (dn ) satisfies the following recursion
dN (iT , y) :=
dn (iT , y)
=
0
∗
((N − (n + 1))h + 1)rh + sup Cn+1
(c̃)
1
0<c̃< h
Z
+((N − (n + 1))h + 1) sup log(1 + α⊤ (e−rh R(z) − 1d ))Q̂Z
n (dz | iT , y)
α∈D
Z
+ dn+1 (iT , y + z)Q̂Z
(4.20)
n (dz | iT , y);
Cn∗ (c̃)
≔
h log(c̃) + (1 + (N − n)h) log(1 − hc̃).
The value of the consumption-investment problem is given by
V0∗ (x0 , iT ) = V0 (x0 , iT , 0) = (1 + T ) log(x0 ) + d0 (iT , 0).
∗
b) For n = 0, . . . , N − 1 let c̃∗n be the maximum point of Cn+1
(·), i.e.
c̃∗n =
1
(N − n)h + 1
∗ be the maximizer of (4.20), i.e.
and let αn
Z
⊤ −rh
Z
∗
log(1 + α (e
R(z) − 1d ))Q̂n (dz | iT , y) .
αn (iT , y) = arg max
α∈D
∗)
Then the optimal consumption-investment strategy is given by (ĉ∗n , α̂n
n∈{0,1,...,N −1} with
∗
α̂n
(iT , z1 , . . . , zn ) :=
∗
αn
(iT ,
n
X
zk );
k=1
ĉ∗n (x, iT , z1 , . . . , zn ) :=
c̃∗n x.
The value function can be separated into two parts: The first part depends on the level of wealth
and the second part depends on the observation and the insider information. Moreover, the
optimal consumed fraction of wealth c̃∗n is only a deterministic function which is increasing in time.
∗ which are invested into the stocks depend
On the other hand, the optimal fractions of wealth α̂n
n
P
on the insider information and through the sum
zk on the history hn = (z1 , . . . , zn ). However,
k=1
∗ does also not depend on the level of wealth. We are not able to present the optimal investment
α̂n
∗ has to be computed using numerical optimization and
strategy in an explicit form. Hence, α̂n
integration procedures.
53
4 Discrete-Time Consumption-Investment Problems
Proof. We show that the structure assumptions (SAN ) are satisfied with
Mn := {v : E → [−∞, ∞) : v(x, iT , y) = (1 + (N − n)h) log(x) + d(iT , y)
o
for a measurable function d with d(iT , y) ≤ b(1 + eb|y|+b|iT | ) for some b ∈ R≥0 ,
∆n := f ∈ Fn with f = (f 1 , f 2 ) : E → A :
1
f 1 (x, iT , y) ≡ f˜1 x for f˜1 ∈ 0,
(independent of x, iT , y),
h
f 2 (x, iT , y) = f˜2 (iT , y) for a measurable function f˜2 : I × Z → D (independent of x) .
Here |x| :=
d
P
k=1
|xk |. We check the assumptions (i) to (iii) of (SAN ):
(i) VN (x, iT , y) = log(x) ∈ MN ;
(ii), (iii)
Let v ∈ Mn+1 . Then
Tn v(x, iT , y) =
sup
x )×D
(c,α)∈(0, h
Z
Z
h log(c) + v(Tx (x, c, α, z), iT , y + z)Q̂n (dz | iT , y) .
Since x > 0 the transformation c̃ = xc is well-defined. We adapt the set of feasible actions
D(x) according to this transformation, i.e.
1
D̃ :=
0,
× D,
h
so that we obtain by plugging in the transition function
Tn v(x, iT , y)
n
= sup
h(log(c̃) + log(x))
(c̃,α)∈D̃
+
Z
o
v(x(1 − hc̃)erh (1 + α⊤ (e−rh R(z) − 1d )), iT , y + z)Q̂Z
(dz
|
i
,
y)
.
T
n
By the definition of v, namely v(x, iT , y) = (1 + (N − (n + 1))h) log(x) + d(iT , y), we get
Tn v(x, iT , y)
=
sup
1 )×D
(c̃,α)∈(0, h
{h(log(c̃) + log(x)) + (1 + (N − (n + 1))h)(log(x) + log(1 − hc̃) + rh)
Z
+ (1 + (N − (n + 1))h) log(1 + α⊤ (e−rh R(z) − 1d ))Q̂Z
n (dz | iT , y)
Z
+ d(iT , y + z)Q̂Z
n (dz | iT , y) } .
We split the supremum within Tn v. Thus
Tn v(x, iT , y)
= (1 + (N − n)h) log(x) + (1 + (N − (n + 1))h)rh
+ sup h log(c̃) + (1 + (N − (n + 1))h) log(1 − hc̃)
1)
c̃∈(0, h
Z
=
54
⊤
log(1 + α (e
+(1 + (N − (n + 1))h) sup
α∈D
Z
+ d(iT , y + z)Q̂Z
n (dz | iT , y)
˜ T , y),
(1 + (N − n)h) log(x) + d(i
−rh
R(z) − 1d ))Q̂Z
n (dz
| iT , y)
(4.21)
4.5 Solutions for Logarithmic and Power Utility Functions
where
˜ T , y)
d(i
sup v c (c̃) + (1 + (N − (n + 1))h) sup v α (α, iT , y) +
≔
1)
c̃∈(0, h
α∈D
+(1 + (N − (n + 1))h)rh +
v c (c̃)
≔
v α (α, iT , y) :=
Z
d(iT , y + z)Q̂Z
n (dz | iT , y)
h log(c̃) + (1 + (N − (n + 1))h) log(1 − hc̃)
Z
log(1 + α⊤ (e−rh R(z) − 1d ))Q̂Z
n (dz | iT , y).
˜ T , y) from above by a function b(1 + eb|iT |+b|y| ) for some b ∈ R≥0
The boundedness of d(i
follows from Lemma 4.4.5. Now we consider the two maximization problems within d˜
separately:
(I) Maximization of v c (·, iT , y) over (0, h1 ):
∂ c
v (c̃, iT , y)
∂c̃
∂2 c
v (c̃, iT , y)
∂c̃2
Setting
∂ c
∂c̃ v (c̃, iT , y) =
h h(1 + (N − (n + 1))h)
−
c̃
1 − hc̃
h h2 (1 + (N − (n + 1))h)
= − 2−
.
c̃
(1 − hc̃)2
=
0 gives us for n = 0, . . . , N − 1 the candidate optimal c̃∗n as
c̃∗n =
1
1 + (N − n)h
2
∂
c
∗
which is independent of (x, iT , y). Since ∂c̃
2 v (c̃, iT , y) < 0 for all c̃ and 0 < c̃n <
candidate optimal c̃∗n is indeed a maximizer of v c over (0, h1 ).
1
h,
the
(II) Maximization of v α (·, iT , y) over D :
Firstly, we show that α 7−→ v α (α, iT , y) is strictly concave over D. For this purpose, let
α, α′ ∈ D with α , α′ (i.e. αk , αk′ for some k ∈ {1, . . . , d}) and let η ∈ (0, 1). Since
1 + (ηα + (1 − η)α′ )(e−rh R(z) − 1)
⊤
−rh
(4.22)
′ ⊤
−rh
= η(1 + α (e
R(z) − 1d )) + (1 − η)(1 + (α ) (e
R(z) − 1d ))
Z
Z
and
1{α⊤ (e−rh R(z)−1d )=(α′ )⊤ (e−rh R(z)−1d )} Q̂Z
n (dz | iT , y) = 0 (Q̂n -zero set)
it follows from the strict concavity of the logarithmic function that α 7−→ v α (α, iT , y) is
also strictly concave over D for all (iT , y). Since we maximize a strictly concave function
∗
which is finite in (iT , y) over a compact set D there always exists a unique maximizer αn
depending on (iT , y); but which is independent of x.
∗)∈∆
Summarizing the two maximization problems, there always exists a maximizer (c∗n , αn
n
∗
∗
with cn = c̃n x so that Tn v = Tn,(c∗n ,α∗n ) v and Tn v is measurable (for the measurability of
Tn v and of the maximizer we refer again to the measurable selection theorem and to Remark
2.3.2 in Bäuerle & Rieder [2011]). Hence, with (4.21) it finally follows that Tn v ∈ Mn .
All in all, Mn and ∆n satisfy (SAN ). From Theorem 4.4.6 it follows:
VN (x, iT , y) =
Vn (x, iT , y) =
log(x)
sup
x )×D
(c,α)∈(0, h
(4.23)
Z
h log(c) + Vn+1 (Tx (x, c, α, z), iT , y + z)Q̂Z
n (dz | iT , y) .
55
4 Discrete-Time Consumption-Investment Problems
We show the statement of the theorem by a backward induction. To this end, let (x, iT , y) ∈ E.
Initial step:
n = N : X since: VN (x, iT , y) = log(x) = log(x) + dN (iT , y)
| {z }
=0
n = N −1 : X
since: We use (4.23) and obtain immediately
n
VN −1 (x, iT , y) = sup
h(log(x) + log(c̃))
(c̃,α)∈D̃
+
Z
o
log x(1 − hc̃)erh (1 + α⊤ (e−rh R(z) − 1d )) Q̂Z
N −1 (dz | iT , y)
=(1 + h) log(x) + rh
∗
(c̃) + sup
+ sup CN
1
0<c̃< h
α∈D
Z
(dz
|
i
,
y)
log 1 + α⊤ (e−rh R(z) − 1d ) Q̂Z
T
N −1
=(1 + h) log(x) + dN −1 (iT , y).
Inductive step: Let Vn be given as in (4.19) (induction hypothesis).
n → n − 1 : From (4.23) we know that
Vn−1 (x, iT , y) = sup
h(log(x) + log(c̃))
(c̃,α)∈D̃
+
Z
Vn (x(1 − hc̃)erh (1 + α⊤ (e−rh R(z) − 1d )), iT , y + z)Q̂Z
n−1 (dz | iT , y) .
With the induction hypothesis we get
h(log(x) + log(c̃))
Vn−1 (x, iT , y) = sup
(c̃,α)∈D̃
+
+
Z
Z
(1 + (N − n)h) log(xerh (1 − hc̃)(1 + α⊤ (e−rh R(z) − 1d )))Q̂Z
n−1 (dz | iT , y)
dn (iT , y + z)Q̂Z
n−1 (dz | iT , y) .
The assertion follows by splitting the supremum
Vn−1 (x, iT , y) =(1 + (N − (n − 1))h) log(x) + (1 + (N − n)h)rh
+ sup {h log(c̃) + (1 + (N − n)h) log(1 − hc̃)}
1
0<c̃< h
Z
⊤
log(1 + α (e
+ (1 + (N − n)h) sup
α∈D
Z
+ dn (iT , y + z)Q̂Z
n−1 (dz | iT , y)
−rh
R(z) − 1d ))Q̂Z
n−1 (dz
| iT , y)
=(1 + (N − (n − 1))h) log(x) + (1 + (N − n)h)rh + sup Cn∗ (c̃)
1
0<c̃< h
log(1 + α⊤ (e−rh R(z) − 1d ))Q̂Z
(dz
|
i
,
y)
+ (1 + (N − n)h) sup
T
n−1
α∈D
Z
+ dn (iT , y + z)Q̂Z
n−1 (dz | iT , y)
Z
=(1 + (N − (n − 1))h) log(x) + dn−1 (iT , y).
56
4.5 Solutions for Logarithmic and Power Utility Functions
Eventually, we obtain from Theorem 4.4.6 part c) together with the two maximization problems
∗
(I) and (II) the optimal consumed fraction of wealth c̃∗n and the optimal fractions of wealth αn
which are invested into the stocks at time tn = nh, namely
c̃∗n
=
∗
αn
(iT , y) =
1
,
(N − n)h + 1
Z
⊤ −rh
Z
log(1 + α (e
R(z) − 1d ))Q̂n (dz | iT , y) .
arg max
α∈D
As aforementioned in Remark 4.4.1 the terminal wealth problem can be solved analogously. Here
we obtain the following result.
Theorem 4.5.2. For the terminal wealth problem it holds:
a) For n = 0, 1, . . . , N the h-insider’s value function can be computed recursively
Vn (x, iT , y) = log(x) + dn (iT , y),
(x, iT , y) ∈ E,
where the sequence (dn ) satisfies the following recursion
dN (iT , y) :=
dn (iT , y)
=
0,
Z
⊤
−rh
R(z) − 1d ))Q̂Z
n (dz
log(1 + α (e
rh + sup
α∈D
Z
+ dn+1 (iT , y + z)Q̂Z
n (dz | iT , y).
| iT , y)
(4.24)
The value of the terminal wealth problem is given by
V0∗ (x0 , iT ) = V0 (x0 , iT , 0) = log(x0 ) + d0 (iT , 0).
∗ be the maximizer of (4.24), i.e.
b) For n = 0, . . . , N − 1 let αn
Z
∗
(dz
|
i
,
y)
.
log(1 + α⊤ (e−rh R(z) − 1d ))Q̂Z
(iT , y) = arg max
αn
T
n
α∈D
Then the optimal fractions of wealth which are invested into the stocks at time tn = nh are
given by
∗
α̂n
(iT , z1 , . . . , zn ) ≔ αn (iT ,
n
X
zk ).
k=1
Here the optimal fractions of wealth which are invested into the stocks are the same as in the
problem with consumption (compare Theorem 4.5.1).
4.5.2 Power Utility Functions
We consider an investor with power utility functions, i.e.
Uc (x) = Up (x) = xγ , 0 < γ < 1,
with dom(Uc ) = dom(Up ) = [0, ∞).
57
4 Discrete-Time Consumption-Investment Problems
Theorem 4.5.3. For the consumption-investment problem it holds:
a) For n = 0, . . . , N the h-insider’s value function can be computed recursively
Vn (x, iT , y) = xγ dn (iT , y),
(x, iT , y) ∈ E,
(4.25)
where the sequence (dn ) satisfies the following recursion
dN (iT , y)
:=
dn (iT , y)
=
with
1;
δ 1−γ
h + dˆn (iT , y)
δ :=(1 − γ)−1 ;
Z
rhγ
ˆ
dn (iT , y) :=e
sup (1 + α⊤ (e−rh R(z) − 1d ))γ dn+1 (iT , y + z)Q̂Z
n (dz | iT , y).
(4.26)
α∈D
The value of the consumption-investment problem is given by
V0∗ (x0 , iT ) = V0 (x0 , iT , 0) = xγ0 d0 (iT , 0).
b) For n = 0, 1, . . . , N − 1 let c̃∗n be given by
c̃∗n (iT , y) = dn (iT , y)−δ
∗ be the maximizer of (4.26), i.e.
and let αn
Z
γ
∗
(dz
|
i
,
y)
.
1 + α⊤ (e−rh R(z) − 1d ) dn+1 (iT , y + z)Q̂Z
(iT , y) = arg max
αn
T
n
α∈D
Then for a wealth x > 0 the optimal consumption-investment strategy is given by
∗ , ĉ∗ )
(α̂n
n n∈{0,1,...,N −1} with
∗
∗
(iT ,
(iT , z1 , . . . , zn ) := αn
α̂n
ĉ∗n (x, iT , z1 , . . . , zn ) := c̃∗n (iT ,
n
X
zk );
k=1
n
X
zk )x.
k=1
∗ = 0.
If x = 0, then the optimal consumed and invested fractions are given by c̃∗n = 0 and α̂n
Remark 4.5.4. Whereas for an initial wealth x0 > 0 the optimal strategy keeps the wealth strictly
positive all the time, the wealth stays always at zero for x0 = 0.
The value function can be written as the product consisting of the power utility of the wealth
and a function which depends on the insider information and the observation. In contrast to
the consumption-investment problem with logarithmic utility functions, the maximization with
respect to the consumption and the investment cannot be separated. Thus, the optimal consumption at time tn = nh depends also on the optimal investment decision (at time tn ). The optimal
58
4.5 Solutions for Logarithmic and Power Utility Functions
consumed fraction of wealth and the optimal investment strategy depend on the insider information and the observation; but do not depend on the level of wealth. We are again not able to
∗)
give the optimal strategy π ∗ = (ĉ∗n , α̂n
n∈{0,...,N −1} in an explicit form. Nevertheless, we reduce
the consumption-investment problem to a stepwise optimization problem where it is enough to
consider Markovian strategies. However, especially for a large number of steps N, it is still hard
∗ ) using numerical optimization and integration procedures. The reason for the
to compute (ĉ∗n , α̂n
∗ ) depends on the
difficulties in numerically determining the strategy is that the decision (ĉ∗n , α̂n
functions dk for all k = n + 1, . . . , N.
Proof. We show that the structure assumptions (SAN ) are satisfied with
M := {v : E → [−∞, ∞) : v(x, iT , y) = xγ d(iT , y)
for a measurable function d with 0 < d(iT , y) ≤ b(1 + eb|y|+b|iT | ) for some b ∈ R>0 } ,
n
∆n := f ∈ Fn with f = (f 1 , f 2 ) : E → A :
1
f 1 (x, iT , y) = f˜1 (iT , y)x for a measurable function f˜1 : I ×Z → [0, ] (independent of x),
h
o
f 2 (x, iT , y) = f˜2 (iT , y) for a measurable function f˜2 : I ×Z → D (independent of x) .
Here |x| :=
d
P
k=1
|xk |. We check the assumptions (i) to (iii) of (SAN ):
(i) VN (x, iT , y) = xγ ∈ M
(ii), (iii)
Let v ∈ M.
Tn v(x, iT , y)
=
sup
(c,α)∈D(x)
=
sup
h log(c) +
x ]×D
(c,α)∈[0, h
Z
v(Tx (x, α, c, z), iT , y + z)Q̂Z
n (dz
| iT , y)
Z
Z
h log(c) + v(Tx (x, α, c, z), iT , y + z)Q̂n (dz | iT , y) .
If x = 0 the only admissible action is (c, α) = (0, 0). Now let x > 0 such that the transformation c̃ = xc is well-defined. We adopt the set of feasible actions according to this
transformation
1
D̃ :=
0,
× D.
h
By plugging in the transition function and using the definition v(x, iT , y) := xγ d(iT , y), we
obtain
Tn v(x, iT , y)
Z
o
n
γ
= sup
hc̃γ xγ + erhγ xγ (1 − hc̃)γ (1 + α⊤ (e−rh R(z) − 1d ) d(iT , y + z)Q̂Z
n (dz | iT , y) .
(c̃,α)∈D̃
We split the supremum
Tn v(x, iT , y)
Z
=xγ sup hc̃γ +(1−hc̃)γ erhγ sup
(1+α⊤ (e−rh R(z)−1d ))γ d(iT , y+z)Q̂Z
n (dz | iT , y)
1]
c̃∈[0, h
˜ T , y),
=x d(i
γ
α∈D
(4.27)
59
4 Discrete-Time Consumption-Investment Problems
where
˜ T , y) :=
d(i
v α (α, iT , y) :=
v c (iT , y) :=
sup
1]
c̃∈[0, h
Z
γ
hc̃ + (1 − hc̃)γ erhγ sup v α (α, iT , y) ;
α∈D
(1 + α⊤ (e−rh R(z) − 1d ))γ d(iT , y + z)Q̂Z
n (dz | iT , y);
hc̃γ + (1 − hc̃)γ erhγ sup v α (α, iT , y).
α∈D
˜ T , y) is positive as well. Furthermore, d(i
˜ T , y) is bounded
Since d(iT , y + z) is positive, d(i
b|y|+b|i
|
T
from above by a function b(1 + e
) for some b ∈ R>0 (compare Lemma 4.4.5). Now
we consider the two maximization problems within d˜ separately:
(I) Maximization of v α (·, iT , y) over D:
Since α 7−→ v α (α, iT , y) is strictly concave (compare proof of Theorem 4.4.6 and use the
strict concavity of xγ , 0 < γ < 1) and we maximize over a compact set D, there always exists
∗ which depends on (i , y); but which is independent of the level of
a unique maximizer αn
T
wealth.
(II) Maximization of v c (·, iT , y) → over [0, h1 ]:
To simplify the notation we define
ṽ α (iT , y) := erhγ sup v α (α, iT , y).
α∈D
In order to compute the maximizer, we firstly determine the partial derivatives of v c w.r.t.
c̃, namely
∂ c
v (c̃, iT , y)
∂c̃
∂2 c
v (c̃, iT , y)
∂c̃2
Setting
∂ c
∂c̃ v (c̃, iT , y) =
= hγc̃γ−1 − hγ(1 − hc̃)γ−1 ṽ α (iT , y)
= hγ(γ − 1)c̃γ−2 + h2 γ(γ − 1)(1 − hc̃)γ−2 ṽ α (iT , y).
0 gives us for (iT , y) ∈ I × Z the candidate optimal c̃∗n as
c̃∗n =
1
(ṽ α (iT , y))−δ
= α
α
−δ
1 + h(ṽ (iT , y))
(ṽ (iT , y))δ + h
(4.28)
with c̃∗n ∈ [0, h1 ] (since ṽ α (iT , y) > 0 for all (iT , y) ∈ I × Z). Moreover, for all c̃ ∈ [0, h1 ] it
holds that
∂2 c
v (c̃, iT , y) = hγ(γ − 1) c̃|γ−2
+ h2 γ(γ − 1) (1 − hc̃)γ−2 ṽ α (iT , y) < 0.
| {z } {z } | {z } |
{z
} | {z }
∂c̃2
<0
c̃∗n
≥0
<0
vc
≥0
>0
1
h]
Therefore,
is indeed a maximizer of
over [0,
which depends on (iT , y); but is
independent of x. Using (4.28) we can simplify the representation of d˜
γ
h
h
˜ T , y) =
γ + 1 − α
ṽ α (iT , y)
d(i
(ṽ (iT , y))δ + h
(ṽ α (iT , y))δ + h
1−γ
=
h + (ṽ α (iT , y))δ
.
Summarizing the two maximization problems (I) and (II), there always exists a maximizer
∗ ) ∈ ∆ with c∗ = c̃∗ x so that T v = T
∗ v. Hence, Tn v is measurable (see
(c∗n , αn
n
n
n,(c∗
n
n
n ,αn )
again Bäuerle & Rieder [2011]). Finally, with (4.27) it follows that Tn v ∈ M.
60
4.5 Solutions for Logarithmic and Power Utility Functions
All in all, M and ∆n satisfy (SAN ). From Theorem 4.4.6 it follows:
VN (x, iT , y) = xγ
Vn (x, iT , y) =
(4.29)
sup
x ]×D
(c,α)∈[0, h
hcγ +
Z
Vn+1 (Tx (x, c, α, z), iT , y + z)Q̂Z
n (dz | iT , y) .
We show the statement by a backward induction. For this purpose, let (x, iT , y) ∈ E.
Initial step:
n=N : X
since: VN (x, iT , y) = xγ = xγ dN (iT , y)
| {z }
=1
n = N −1 :
since: From (4.29) we know that
X
VN −1 (x, iT , y)
Z
n
o
=
sup
hc̃γ xγ + (x(1 − hc̃)erh (1 + α⊤ (e−rh R(z) − 1d )))γ Q̂Z
(dz
|
i
,
y)
T
N −1
1 ]×D
(c̃,α)∈[0, h
Z
n
o
γ
γ rhγ
=x
sup
hc̃ +(1−hc̃) e
(1+α⊤ (e−rh R(z)−1d ))γ dN (iT , y+z) Q̂Z
(dz
|
i
,
y)
.
T
|
{z
} N −1
(c̃,α)∈[0, 1 ]×D
γ
h
=1
Using (4.26) and (4.28) we obtain
n
o
1−γ
VN −1 (x, iT , y) =xγ sup hc̃γ + (1 − hc̃)γ dˆN −1 (iT , y) = xγ h + (dˆN −1 (iT , y))δ
1
0≤c̃≤ h
=xγ dN −1 (iT , y).
Inductive step: Let Vn be given as in (4.25) (induction hypothesis).
n → n − 1 : From (4.29) we know that
Vn−1 (x, iT , y)
Z
n
o
=
sup
hxγ c̃γ + Vn (x(1 − hc̃)erh (1 + α⊤ (e−rh R(z) − 1d )), iT , y + z)Q̂Z
n−1 (dz | iT , y) .
1 ]×D
(c̃,α)∈[0, h
By the induction hypothesis we get
Vn−1 (x, iT , y)
=xγ
sup
1 ]×D
(c̃,α)∈[0, h
=xγ sup
1
0≤c̃≤ h
Z
n
o
hc̃γ + (1 − hc̃)γ erhγ (1 + α⊤ (e−rh R(z) − 1d ))γ dn (iT , y+z)Q̂Z
(dz
|
i
,
y)
T
n−1
n
o
hc̃γ + (1 − hc̃γ )dˆn−1 (iT , y) .
Finally, we plug in the optimal consumption (4.28) and the assertion follows
1−γ
Vn−1 (x, iT , y) =xγ h + (dˆn−1 (iT , y))δ
= xγ dn−1 (iT , y).
Eventually, we obtain from Theorem 4.4.6 part c) together with the maximization problems (I)
∗ which
and (II) the optimal consumed fraction of wealth c̃∗n and the optimal fractions of wealth αn
are invested into the stocks at time tn = nh, namely
Z
γ
∗
⊤ −rh
Z
αn (iT , y) = arg max
1 + α (e
R(z) − 1d ) dn+1 (iT , y + z)Q̂n (dz | iT , y) ;
α∈D
c̃∗n (iT , y)
= dn (iT , y)−δ .
61
4 Discrete-Time Consumption-Investment Problems
Analogously, we obtain the following result:
Theorem 4.5.5. For the terminal wealth problem it holds:
a) For n = 0, . . . , N the h-insider’s value function can be computed recursively
Vn (x, iT , y) = xγ · dn (iT , y),
(x, iT , y) ∈ E,
where the sequence (dn ) satisfies the following recursion
dN (iT , y) := 1,
dn (iT , y) = erhγ sup
α∈D
Z
(1 + α⊤ (e−rh R(z) − 1))γ dn+1 (iT , y + z)Q̂Z
(dz
|
i
,
y)
.
T
n
(4.30)
The value of the terminal wealth problem is given by
V0∗ (x0 , iT ) = V0 (x0 , iT , 0) = xγ0 d0 (iT , 0).
∗ be the maximizer of (4.30), i.e.
b) For n = 0, 1, . . . , N − 1 let αn
Z
∗
⊤ −rh
γ
Z
αn (iT , y) = arg max
(1 + α (e
R(z) − 1d )) dn+1 (iT , y + z)Q̂n (dz | iT , y) .
α∈D
Then the optimal fractions of wealth which are invested into the stocks at time tn = nh are
given by
∗
∗
α̂n
(iT , z1 , . . . , zn ) ≔ αn
(iT ,
n
X
zk ).
k=1
The optimal fractions which are invested into the stocks at time tn = nh depend on dn+1 . For the
terminal wealth and the consumption-investment problem the sequences (dn ) are different. Hence,
in contrast to logarithmic utility functions, the optimal investment strategies in the problem with
and without consumption are not identical.
4.6 Properties of the Portfolio Value and the Optimal Strategy
In this section we analyze the behavior of the results obtained in the previous sections with respect to changes in some model parameters. We illustrate some results with a numerical example.
In our numerical example we always consider a financial market with one stock (d = 1). Unless
stated otherwise, we choose the following parameters:
As often used in the literature (e.g. Bäuerle et al. [2012]) we choose the interest rate and the
volatility as r = 0.02 and σ = 0.3, the unknown drift is sampled from a normal distribution with
expectation λ0 = 0.3 and variance Σ0 = 0.5; concerning our insider information we select a = 0.5
and Σε = 1; the remaining parameters are chosen as γ = 0.6, x0 = 1, N = 1, h = 1.
With this choice of parameters Assumption 4.3.24 is satisfied. In general we choose here for
all three types of insider information the same convex factor a and the same noise Σε . Since
N = 1, there is only one time point t0 = 0, where the investor decides about the consumption and
the investment. The implementation is done in MATLAB using the numerical integration and
optimization toolbox.
62
4.6 Properties of the Portfolio Value and the Optimal Strategy
4.6.1 The Value of Information
So far, we considered the ex-post value - ex-post in the sense that the value of the consumptioninvestment problems depend explicitly on the insider information. Now we define the so-called
ex-ante value.
Definition 4.6.1. The ex-ante value is the expectation of the ex-post value V0 (x0 , IT , 0) with
respect to the insider information
Z
IT
V0 (x0 , 0) := V0 (x0 , iT , 0)QI0T (diT ),
where QI0T denotes the distribution of the insider information IT .
As for instance in Amendinger et al. [2003] and in Hansen [2013] we define the value of the insider
information IT as the certainty equivalent (CEIT ) - also known as indifference price. The latter
is the amount of money which an investor needs additionally to his initial wealth at time t0 = 0 in
order to be indifferent between having insider information IT and having this additional amount
of money. By setting the convex factor to a = 0 we obtain the value function of a regular investor
from the insider optimization problem. We denote the value function of a regular investor by
Vna=0 (x0 , y).
The value function Vn (x, iT , y) and therefore also the sequence (dn ) are different for the three
types of insider information. If we consider a specific type of insider information, we write VnS , dS
n
λ λ
for stock price, VnW , dW
n for Brownian and Vn , dn for drift information.
Definition 4.6.2. The certainty equivalent of the insider information IT is defined as the solution
CEIT of the following equation
V0a=0 (x0 + CEIT , 0) = V0IT (x0 , 0).
An insider is indifferent between the insider information ITX and the insider information ITY if
the corresponding ex-ante values are equal. In what follows, we define the certainty equivalent
between the insider information ITX and ITY as the amount of money which makes an insider
indifferent between having insider information ITX and having insider information ITY .
Definition 4.6.3. The certainty equivalent between the insider information ITX and ITY is defined
as the solution CEXY of the following equation
V0X (x0 + CEXY , 0) = V0Y (x0 , 0),
where X, Y ∈ {S, W, λ} stand for stock price, Brownian or drift information.
Properties of the Conditional Distribution of the Insider Information
We introduce some notations and state some properties of the distribution of the insider information. We formulate these properties in Lemma 4.6.4 to Lemma 4.6.6.
For a = 0 the conditional distributions Q̂Z
n (· | iT , yn ), µ̂n (· | iT , yn ) and µn (· | iT , z1 , . . . , zn ) do not
depend on the insider information. Therefore, we denote them shortly by Q̂Z
n (· | yn ), µ̂n (· | yn )
and µn (· | z1 , . . . , zn ).
63
4 Discrete-Time Consumption-Investment Problems
Lemma 4.6.4. The (regular) conditional distribution QInT of IT given the unknown drift λ = θ
and the past observations Hn = hn follows a multivariate normal distribution which depends only
n
P
through
zk on the past observations. Hence, there exists a transition kernel (Q̄InT ) from Θ × Z
k=1
to I such that
QInT (· | θ, hn ) = Q̄InT (· | θ,
for n = 0, . . . , N − 1 with
QI0T (·
| θ, h0 ) :=
QI0T (·
n
X
zk )
k=1
| θ) = Q̄I0T (·
| θ, 0).
Proof. For the distribution QInT (· | θ, hn ) we refer to the proofs of Theorem 4.3.2, Theorem 4.3.4
and Theorem 4.3.6.
For all C ∈ B(I) we define
Q̂InT (C | y) :=
Z
Q̄InT (C | θ, y)µ̂n (dθ | y).
(4.31)
Then the next lemma can be shown similarly to Lemma 4.3.22.
Lemma 4.6.5. Q̂InT (· | y) follows a multivariate normal distribution with expectation and covariance matrix given in Table 4.4.
Type of
insider
information
Expectation
Covariance
matrix
IT = ITS
a(nhΣ0 + Id )−1((Id + N hΣ0 )y+(N − n)hλ0 )
Kn + a2 (N − n)2 h2 (nhΣ0 + Id )−1 Σ0
IT = ITW
a(nhΣ0 + Id )−1 (y − nhλ0 )
Kn + a2 n2 h2 (nhΣ0 + Id )−1 Σ0
IT = ITλ
a(nhΣ0 + Id )−1 (Σ0 y + λ0 )
(1 − a)2 Σε + a2 (nhΣ0 + Id )−1 Σ0 .
Table 4.4: Expectation and covariance matrix of Q̂InT (· | y)
The conditional distribution of IT given the history hn depends on the past observations only
through its sum. Therefore, by the tower property of the conditional expectation it follows:
Lemma 4.6.6. Let B ∈ B(I) and let C ∈ B(Z). Then it holds:
Z Z
Z Z
Z
IT
T
Q̂n (dz | iT , y)Q̂n (diT | y) =
Q̂In+1
(diT | y + z)Q̂Z
n (dz | y).
B C
C B
Logarithmic Utility Functions
Firstly, we consider an investor with logarithmic utility functions, i.e. Uc (x) = Up (x) = log(x).
Hence, for (x, iT , y) ∈ E the value function is given by
Vn (x, iT , y) = (1 + (N − n)h) log(x) + dn (iT , y),
with dn defined in Theorem 4.5.1 (recursion). If the investor does not have insider information,
we denote latter by
da=0
n (y).
It follows directly:
64
4.6 Properties of the Portfolio Value and the Optimal Strategy
Theorem 4.6.7.
a) The certainty equivalent is given by
"
CEIT = x0 exp
R
d0 (iT , 0)QI0T (diT ) − da=0
(0)
0
Nh+1
!
#
−1 .
b) The certainty equivalent between the insider information ITX and ITY is given by
CEXY = x0 exp
R
R
X
dY0 (iT , 0)QY0 (diT ) − dX
0 (iT , 0)Q0 (diT )
−1 ,
Nh+1
where X, Y ∈ {S, W, λ} stand for stock price, Brownian or drift information.
CEI
The fraction of the initial wealth which makes an investor indifferent, namely x0 T , does not
depend on the level of the initial wealth. This is a consequence of the constant relative risk aversion
of the logarithmic utility function. Furthermore, from the following theorem we conclude that
the certainty equivalent is always non-negative. Thus, in order to achieve the same ex-ante value
the initial wealth of a regular investor can never be less than that of an insider (x0 + CEIT ≥ x0 ).
Since the set of admissible strategies of a regular investor is a subset of the set of admissible
strategies of an insider, this result seems to be intuitive.
Theorem 4.6.8. For n = 0, . . . , N it holds:
Z
dn (iT , y)Q̂InT (diT | y) ≥ da=0
n (y)
for all y ∈ Z.
Hence, the certainty equivalent CEIT is always non-negative.
Proof. We show the statement by a backward induction. For this purpose, let (iT , y) ∈ I × Z.
Initial step:
n = N : X since: dN (iT , y) = 0 = da=0
N (y);
n = N − 1 : X since: By "interchanging" the expectation and the supremum we obtain the
following inequality
Z
dN −1 (iT , y)Q̂INT−1 (diT | y)
Z
Z
IT
∗
=rh + sup CN
(c̃) + sup log(1 + α⊤ (e−rh R(z) − 1d ))Q̂Z
N −1 (dz | iT , y)Q̂N −1 (diT | y)
1
0<c̃< h
α∈D
∗
(c̃) + sup
≥rh + sup CN
"
∗
=rh + sup CN
(c̃) + sup
Z
1
0<c̃< h
1
0<c̃< h
α∈D
α∈D
IT
log(1 + α⊤ (e−rh R(z) − 1d ))Q̂Z
N −1 (dz | iT , y)Q̂N −1 (diT | y)
log(1 + α⊤ (e−rh R(z) − 1d ))Q̂Z
N −1 (dz | y)
=da=0
N −1 (y),
where we use the tower property of the conditional expectation in the second equality.
R
T
Inductive step: Let dn+1 (iT , y)Q̂In+1
(diT | y) ≥ da=0
n+1 (y) for all y ∈ Z (induction hypothesis).
n + 1 → n : Let y ∈ Z. Analogously to the initial step we obtain by "interchanging" the expectation
65
4 Discrete-Time Consumption-Investment Problems
and the supremum the following inequality
Z
dn (iT , y)Q̂InT (diT | y)
∗
(c̃) +
=((N − (n + 1))h + 1)rh + sup Cn+1
1
0<c̃< h
Z
"
IT
dn+1 (iT , y + z)Q̂Z
n (dz | iT , y)Q̂n (diT | y)
Z
IT
log(1 + α⊤ (e−rh R(z) − 1d ))Q̂Z
n (dz | iT , y)Q̂n (diT | y)
"
∗
IT
dn+1 (iT , y + z)Q̂Z
≥((N − (n + 1))h + 1)rh + sup Cn+1 (c̃) +
n (dz | iT , y)Q̂n (diT | y)
+ ((N − (n + 1))h + 1)
sup
α∈D
1
0<c̃< h
"
IT
log(1 + α⊤ (e−rh R(z) − 1d ))Q̂Z
n (dz | iT , y)Q̂n (diT | y)
"
∗
T
(diT | y + z)Q̂Z
dn+1 (iT , y + z)Q̂In+1
=((N − (n + 1))h + 1)rh + sup Cn+1 (c̃) +
n (dz | y)
+ ((N − (n + 1))h + 1) sup
α∈D
1
0<c̃< h
+ ((N − (n + 1))h + 1) sup
α∈D
Z
log(1 + α⊤ (e−rh R(z) − 1d ))Q̂Z
n (dz | y),
where we use Lemma 4.6.6 in order to get the last equality. Then by the induction hypothesis we
obtain the first assertion
"
∗
T
((N − (n + 1))h + 1)rh + sup Cn+1 (c̃) +
(diT | y + z)Q̂Z
dn+1 (iT , y + z)Q̂In+1
n (dz | y)
1
0<c̃< h
Z
log(1 + α⊤ (e−rh R(z) − 1d ))Q̂Z
n (dz | y)
Z
∗
Z
≥((N − (n + 1))h + 1)rh + sup Cn+1
(c̃) + da=0
n+1 (y + z)Q̂n (dz | y)
+ ((N − (n + 1))h + 1) sup
α∈D
1
0<c̃< h
+ ((N − (n + 1))h + 1) sup
α∈D
=da=0
n (y).
Z
log(1 + α⊤ (e−rh R(z) − 1d ))Q̂Z
n (dz | y)
For n = 0 and with Q̂I0T (· | 0) = QI0T (·) we get
Z
d0 (iT , 0)QI0T (diT ) ≥ da=0
(0).
0
Consequently, the second statement follows by Theorem 4.6.7.
Under some assumptions on the initial parameters, the h-insider is indifferent between the insider
information ITX and ITY (the variables X, Y ∈ {S, W, λ} stand for stock price, Brownian or drift
information). In the following theorem we allow different convex factors for the insider information
ITX and ITY . Nevertheless, we still suppose the same covariance matrix Σε (of the noises). We
X
denote by ΣX
a′ the modified covariance matrix associated to the insider information IT with
′
convex factor a ∈ [0, 1).
Theorem 4.6.9. Let Σ0 and Σε be simultaneously diagonalizable. Moreover, let a′ ∈ [0, 1) be the
Y
convex factor of ITX and let a ∈ [0, 1) be the convex factor of ITY . If ΣX
a′ = Σa , then the investor
X
Y
is indifferent between the insider information IT and IT (i.e. CEXY = 0).
66
4.6 Properties of the Portfolio Value and the Optimal Strategy
W
Remark 4.6.10. In the one-dimensional case (d = 1) it holds that ΣS
a ≤ Σa ≤ Σ0 (with the same
S
W
convex factor a), where the equality Σa = Σa is only satisfied if there is no drift uncertainty (i.e.
W Z has in general
Σ0 = 0). A smaller value of ΣIaT implies a smaller variance of Q̂Z
n . Hence, Q̂n
SZ
more variability than Q̂n .
As ΣIaT does not have an economic meaning, the conditions in Theorem 4.6.9 under which an
investor attributes the same value to the different types of insider information appear to be only
Y
mathematical. However, for d = 1 it is straightforward to verify that the condition ΣX
a′ = Σa is
satisfied if and only if
Corr(ITX , log(ST )) = Corr(ITY , log(ST )).
Hence, an investor is indifferent between the different types of insider information if the correlations between the terminal logarithmic stock price and each type of insider information are
identical.
Proof. We prove the statement only for ITX = ITW and ITY = ITλ . Firstly, we introduce some
notations
Z
• Here q̂nW Z (· | iT , yn ) (respectively q̂nλZ (· | iT , yn )) denotes the Lebesgue density of Q̂W
n (· |
λZ
iT , yn ) (respectively Q̂n (· | iT , yn ));
• Recall the distributions of the Brownian and the drift information
Qλ
0 (·)
QW
0 (·)
= N (aλ0 , a2 Σ0 + (1 − a)2 Σε );
2
= N (0, a′ T Id + (1 − a′ )2 Σε ).
λ
We denote by q0W (·) (respectively q0λ (·)) the Lebesgue density of QW
0 (·) (respectively Q0 (·)).
In order to prove the statement, namely
Z
Z
λ
λ
W
d0 (iT , 0)Q0 (diT ) = dW
0 (iT , 0)Q0 (diT ),
it is enough to show that for n = 0, . . . , N − 1 and with yn =
n
P
(4.32)
zk it holds
k=1
Z
Z
λZ
f (α, zn+1 )q̂nλZ (zn+1 | iT , yn )dzn+1 q̂n−1
. . . sup
(zn | iT , yn−1 )dzn . . .
α∈D
| {z }
Z Z
n−times
=
Z Z
...
Z
sup
α∈D
Z
. . . q̂0λZ (z1 | iT , 0)dz1 q0λ (iT )diT
WZ
WZ
f (α, zn+1 )q̂n (zn+1 | iT , yn )dzn+1 q̂n−1
(zn | iT , yn−1 )dzn . . .
. . . q̂0W Z (z1 | iT , 0)dz1 q0W (iT )diT ,
(4.33)
where
f (α, z) := log(1 + α⊤ (e−rh R(z) − 1d )).
The matrices Σ0 and Σε are simultaneously diagonalizable, i.e. Σ0 = OD0 O⊤ and Σε = ODε O⊤
with positive definite diagonal matrices D0 , Dε and an orthogonal matrix O. Let σλ be the (unique)
67
4 Discrete-Time Consumption-Investment Problems
positive definite square root of a2 Σ0 + (1 − a)2 Σε and let σW be the (unique) positive definite
2
square root of a′ T Id + (1 − a′ )2 Σε , i.e.
σλ
σW
1
:= O[a2 D0 + (1 − a)2 Dε ] 2 O⊤ ;
1
2
:= O[a′ T Id + (1 − a′ )2 Dε ] 2 O⊤ ,
1
where [A] 2 denotes the square root of a matrix A (which we get here by taking the square roots
of the diagonal elements). Now we follow the idea of Hansen [2013] Proposition 3 and define
Φ := σW σλ−1 (ITλ − aλ0 ).
d
d
2
Then Φ = ITW = N (0, a′ T Id + (1 − a′ )2 Σε ). Therefore, we consider the transformation
φ := σW σλ−1 (iT − aλ0 ),
(4.34)
where the Jacobian of the latter is given by σW σλ−1 . Consequently, we obtain
q0W (φ)dφ
1
=
d
2
1
2
exp(− (φ⊤ (a′ T Id + (1 − a′ )2 Σε )−1 φ))dφ
2
q
det (a′ 2 T Id + (1 − a′ )2 Σε )
det (σW σλ−1 )
1
−1
⊤ −1
−1
((i
−
aλ
)
σ
σ
(σ
σ
)
σ
σ
(i
−
aλ
))
diT
exp
−
=
0
0
T
W
W W
W λ
T
d
λ
2
(2π) 2 det σW
1
1
⊤ 2
2
−1
exp − ((iT − aλ0 ) (a Σ0 + (1 − a) Σε ) (iT − aλ0 )) diT
=
d p
2
(2π) 2 det(a2 Σ0 + (1 − a)2 Σε )
(2π)
=q0λ (iT )diT .
(4.35)
Since Σ0 and Σε are simultaneously diagonalizable, the (unique) positive definite square root of
Σ0 − Σλ
a exists and is given by
1
1
1
2
2
−1
2
Σ0 ] 2 = aOD0 [(a2 D0 + (1 − a)2 Dε )−1 ] 2 O⊤ .
[Σ0 − Σλ
a ] = a[Σ0 (a Σ0 + (1 − a) Σε )
λ
Thus, we get with ΣW
a′ = Σa (by assumption) and with the transformation (4.34) that
Cλλ λ0 + C λ iT
1
2 −1
= λ0 + [Σ0 − Σλ
a ] σλ (iT − aλ0 )
1
2 −1
= λ0 + [Σ0 − ΣW
a′ ] σ W φ
= CλW λ0 + C W φ.
68
(4.36)
4.6 Properties of the Portfolio Value and the Optimal Strategy
Let us use the abbreviation VnIT := Id + nhΣIaT . Then by (4.34) and (4.36) we obtain
q̂nλZ (z | iT , yn )
p
det Vnλ
q
exp
=
d
λ
(2π) 2 h det Vn+1
=
p
det VnW
q
exp
d
W
(2π) 2 h det Vn+1
=q̂nW Z (z | φ, yn ).
−
⊤
1 z − (Vnλ )−1 (Cλλ λ0 + C λ iT + Σλ
a yn )
2h
λ
× (Vn+1
)−1 Vnλ
−
z − (Vnλ )−1 (Cλλ λ0 + C λ iT
+ Σλ
a yn )
!
⊤
1 z − (VnW )−1 (CλW λ0 + C W φ + ΣW
a yn )
2h
W −1 W
× (Vn+1
) Vn
z − (VnW )−1 (CλW λ0 + C W φ + ΣW
a yn )
!
(4.37)
Finally, the assertion (4.33) follows by (4.37) together with (4.35). The proof of the indifference
between the other types of insider information follows the same lines. We only need to adapt the
transformation (4.34) according to the considered type of insider information:
• If ITX = ITS and ITY = ITλ , then we consider the transformation
φ := a′ λ0 T + σS σλ−1 (iT − aλ0 ),
2
2
where σS and σλ are the positive definite square roots of a′ T 2 Σ0 + a′ T Id + (1 − a′ )2 Σε
and of a2 Σ0 + (1 − a)2 Σε ;
• If ITX = ITS and ITY = ITW , then we consider the transformation
−1
iT ,
φ := a′ λ0 T + σS σW
2
2
where σS and σW are the positive definite square roots of a′ T 2 Σ0 + a′ T Id + (1 − a′ )2 Σε
and of K0 .
Certainty Equivalent in the Numerical Example (Logarithmic Utility)
Figure 4.3 and Figure 4.4
In Figure 4.3 we plot the certainty equivalent against the convex factor a for different values of
Σ0 . This numerical example confirms that the certainty equivalent is always non-negative. This
is a reasonable fact since the investor will not trade on the basis of the insider information if
there is not the possibility of some extra profit. A convex factor of a = 0 corresponds to the case
without insider information. Hence, the certainty equivalent is here zero. The insider information is a convex combination consisting of exact information and the noise. Loosely speaking,
the influence of the noise decreases when a increases. Therefore, it seems to be intuitive that
the certainty equivalent increases in a: The more exact the insider information, the more money
the investor is willing to pay in order to have the additional information. The variance Σ0 of
the unknown drift determines the public uncertainty. The first row of Figure 4.4 shows the
behavior of the indifference price with respect to changes in the public uncertainty more precisely
(for λ0 = −0.3 and λ0 = 0.3 with a fixed convex factor a = 0.5). For a public uncertainty close
69
4 Discrete-Time Consumption-Investment Problems
Σ0 = 0.5
Σ0 = 0.01
CES
0.15
0.15
CEW
CEIT
CEIT
CEλ
0.1
0.05
0.1
0.05
0
0
0.5
0
0.5
0
1
a
Σ0 = 1
Σ0 = 10
0.15
0.15
0.1
0.1
CEIT
CEIT
1
a
0.05
0.05
0
0
0.5
0
a
1
0.5
0
1
a
Figure 4.3: Certainty equivalent in dependence on a for different Σ0
(Note that on the top left the red curve is (almost) hidden by the green curve and
on the bottom left the green curve is hidden by the blue curve)
to zero the investor is not willing to pay much money in order to have additional information
about the unknown drift. Here, especially for a small value of the convex factor, the drift information does not provide (useful) additional information. However, when the precision of the
unknown drift decreases (Σ0 increases) it becomes more difficult to make the "right" investment
decision. Additional information about the unknown drift increases the chance of making the
"right" decision. Hence, drift information and stock price information become more and more
valuable. On the other hand, the value of Brownian information is for λ0 = 0.3 decreasing in
the public uncertainty. That may result from the fact that the less the uncertainty is the better
the Brownian information describes the terminal stock price. Nevertheless, we observe that all
three types of insider information have a higher value for the positive λ0 than for the negative λ0 .
For Σ0 = 0.01 (for all a) the Brownian and the stock price certainty equivalent are almost equal,
i.e. the investor is here willing to pay approximately the same amount of money for additional
information about the terminal stock price or about the terminal Brownian motion. This makes
perfectly sense since without drift uncertainty Brownian and stock price information represent
exactly the same information (σ(ITS ) = σ(ITW )). For Σ0 = 1 we see that for all a the Brownian
certainty equivalent corresponds to the drift certainty equivalent. Thus, the investor is indifferent
between drift and Brownian information. This confirms the theoretical result of Theorem 4.6.9
as in this example the correlations between the logarithmic terminal stock price and each of the
70
4.6 Properties of the Portfolio Value and the Optimal Strategy
two types of insider information are identical. In the second row of Figure 4.4 we plot the certainty equivalent between the insider information ITX and ITY . The amount of money which the
investor is willing to pay in order to have stock price instead of Brownian information increases
in Σ0 . The intuition behind is that in contrast to Brownian information stock price information
provides also information about the unknown drift. Moreover, it is interesting to observe that
the certainty equivalent between drift and Brownian information becomes negative: The effect of
drift uncertainty seems to outweigh the effect of additional information about the terminal value
of the Brownian motion (for Σ0 > 1).
λ0 = 0.3
0.06
0.06
0.04
0.04
CEIT
CEIT
λ0 = −0.3
0.02
0.02
0
0
−0.02
−0.02
CES
CEW
−0.04
0
1
0.5
1.5
−0.04
2
CEλ
0
Σ0
λ0 = −0.3
CEXY
CEXY
CEλW
0.02
−0.02
−0.02
0.5
1
2
0.02
0
0
1.5
0.04
0
−0.04
2
0.06
CEλS
0.04
1.5
Σ0
λ0 = 0.3
CEW S
0.06
1
0.5
1.5
2
−0.04
0
0.5
Σ0
1
Σ0
Figure 4.4: Certainty equivalent in dependence on Σ0 for different λ0
Optimal Strategy in the Numerical Example (Logarithmic Utility)
Figure 4.5 and Figure 4.6
The optimal investment strategy depends explicitly on the insider information. Since ITS , ITW , ITλ
have not the same distributions, it seems slightly unfair to compare the optimal investment
strategies of the different types of insider information for the same realized value iT of IT ∈
{ITS , ITW , ITλ }. Therefore, we look in Figure 4.5 at the average optimal investment, namely
Z
ᾱ0∗ := α0∗ (iT , 0)QI0T (diT ).
As y represents the past observations it is convenient to set here y = 0. We plot the average
optimal strategy against the public uncertainty Σ0 for different values of the convex factor a and
71
4 Discrete-Time Consumption-Investment Problems
λ0 = −0.3, a = 0.25
λ0 = −0.3, a = 0.75
1
1
Stock price
Brownian
Drift
Without
0.8
0.6
α∗0
α∗0
0.6
0.8
0.4
0.2
0.4
0.2
0
0
0
1
0.5
1.5
2
0
1
0.5
1.5
2
Σ0
λ0 = 0.3, a = 0.75
1
1
0.8
0.8
0.6
0.6
α∗0
α∗0
Σ0
λ0 = 0.3, a = 0.25
0.4
0.2
0.4
0.2
0
0
0
1
0.5
Σ0
1.5
2
0
1
0.5
1.5
2
Σ0
Figure 4.5: Average optimal strategy in dependence on Σ0 for different values of λ0 and a
different values of λ0 . The same plot contains the optimal strategy of a regular investor in order
to see the extent to which the investor reduces or increases his investment into the risky asset
when having additional information. The average optimal strategies differ significantly from the
optimal strategy without insider information (yellow line). This indicates that the insider makes
his decisions on the basis of the additional information. For a public uncertainty going to zero
the average strategy with Brownian information tends to the average strategy with stock price
information. On the other hand, the average strategy with drift information tends to the optimal
strategy without insider information. As aforementioned, this makes totally sense as for Σ0 = 0
drift information is useless and the other two types of insider information represent the same
information. In the first row of Figure 4.5 we keep the initial expectation at λ0 = −0.3 and
consider convex factors of a = 0.25 and a = 0.75. The average optimal strategies are increasing
in the public uncertainty. This seems to be counterintuitive. It might be thought that the
investor tends to reduce the investment into the risky asset when the public uncertainty increases.
Moreover, the investor with stock price information invests more into the stock as the investor
with Brownian information, drift information and as the investor without additional information.
A higher value of λ0 let the investor expect a higher future stock price. Thus, it is reasonable that
the investor invests more into the risky asset when we enhance λ0 to 0.3. The effect of changes in
Σ0 seem now to be reversed: The average strategies decrease in Σ0 and the regular investor invests
more into the risky asset than an investor with insider information (on average). Furthermore,
Figure 4.5 suggests that the changes in the average strategies decrease in Σ0 (Σ0 ∈ [0, 2]): The
72
4.6 Properties of the Portfolio Value and the Optimal Strategy
higher the public uncertainty is, the less is the change in the average strategies for all three types
of insider information. This effect seems to be even more enhanced when we increase the convex
factor. An economic intuition is that when the public uncertainty becomes higher the investor
trusts more in his insider information. Consequently, for Σ0 big enough we observe only small
reactions towards changes in Σ0 . In Figure 4.6 we plot the average optimal strategy against the
time lag h, that is
∗
ᾱn
=
Z
∗
αn
(iT , y)Q̂InT (diT | y).
We set here the investment horizon to T = 10 and fix n = 9. Moreover, we choose the convex
factor a = 0.25 (remaining parameters chosen according to the example on page 62, Assumption
Without insider information
Stock price information
1
1
y=1
0.8
y = 0.25
0.6
0.6
y=0
∗
ᾱn
∗
ᾱn
0.8
y = 0.5
y = −0.5
0.4
0.2
0.4
0.2
0
0
0
0.5
1
0
h
1
h
Brownian information
Drift information
1
1
0.8
0.8
0.6
0.6
∗
ᾱn
∗
ᾱn
0.5
0.4
0.2
0.4
0.2
0
0
0
0.5
h
1
0
0.5
1
h
Figure 4.6: Average optimal strategy in dependence on the time lag h for different realized values
of y
4.3.24 is satisfied). Since we fix the investment horizon the total number of trading time points
N increases when h decreases. Due to the fixed n an increasing h also means that the investment
horizon comes closer. The four plots show the average optimal strategies for the different types
of information: without additional information, with stock price, Brownian and drift information.
Each plot contains the average optimal strategy for different realized values of the observation y
73
4 Discrete-Time Consumption-Investment Problems
(for the dotted lines we refer to Chapter 6). Note that for t = nh it holds
log(Snk ) =
d
X
j=1
1 2
nh.
σkj Ynk − σkj
2
For d = 1 the past stock price is obviously increasing in the observation y. A relatively high past
stock price allows the investor to be more optimistic. Hence, it is intuitively plausible that we
observe the (average) optimal investment strategies to be increasing in y. However, as we see in
the following subsection this is not in general true in case of Brownian and stock price information.
Moreover, we observe that the optimal strategy without insider information is decreasing in h.
This seems to be reasonable: The smaller h, the faster the investor is able to react to changes in
the stock price. Consequently, a high stock position becomes less risky. For drift and Brownian
information we observe a quite similar reaction towards changes in h and y. However, it seems that
for a negative value of y the insider information gives the investor reasons to be more optimistic
such that the investor invests also for a negative y into the stock. If the investor has stock price
information, then the behavior of the average strategy for varying y and h changes substantially
relatively to the strategy without insider information. For a negative value of y it seems that
the average strategy is increasing in h (for h big enough). Although counterintuitive at first, a
greater h also means that the investment horizon comes closer and the information about the
terminal stock price may become more exact and more helpful in the decision making.
Power Utility Functions
We consider an investor with power utility functions, i.e. Uc (x) = Up (x) = xγ , 0 < γ < 1. Hence,
for (x, iT , y) ∈ E the value function is given by
Vn (x, iT , y) = xγ dn (iT , y),
with dn defined in Theorem 4.5.3 (recursion). If the investor does not have insider information,
we denote the recursion by
da=0
n (y).
It follows directly:
Theorem 4.6.11.
a) The certainty equivalent is given by


! γ1
R
IT
(di
)
d
(i
,
0)Q
0 T
T
0
CEIT = x0 
− 1 .
da=0
(0)
0
b) The certainty equivalent between the insider information ITX and ITY is given by


1
R Y
Y (di ) γ
d
(i
,
0)Q
T
0 T
0
CEXY = x0  R X
− 1 ,
d0 (iT , 0)QX
0 (diT )
where X, Y stand for stock price, Brownian and drift information.
The power utility function belongs also to the class of CRRA utility functions. Thus, it comes as
CEI
no surprise that x0 T does not depend on the initial wealth.
74
4.6 Properties of the Portfolio Value and the Optimal Strategy
Theorem 4.6.12. For n = 0, . . . , N it holds:
Z
dn (iT , y)Q̂InT (diT | y) ≥ da=0
n (y)
for all y ∈ Z.
Hence, the certainty equivalent CEIT is always non-negative.
Proof. We show the statement by a backward induction. For this purpose, let (iT , y) ∈ I × Z.
Initial step:
n = N : X since: dN (iT , y) = 1 = da=0
N (y);
n = N − 1 : X since: Due to Theorem 4.5.3 we can write dN −1 (iT , y) as follows
Z
dN −1 (iT , y)Q̂INT−1 (diT
| y) =
=
Z
Z
(h + (dˆN −1 (iT , y))δ )1−γ Q̂INT−1 (diT | y)
sup {hc̃γ + (1 − hc̃)γ dˆN −1 (iT , y)}Q̂INT−1 (diT | y).
1
0≤c̃≤ h
By "interchanging" the expectation and the supremum over c̃ we obtain the following inequality
Z
dN −1 (iT , y)Q̂INT−1 (diT
Z
IT
γ
γ
ˆ
| y) ≥ sup
hc̃ + (1 − hc̃)
dN −1 (iT , y)Q̂N −1 (diT | y) .
1
0≤c̃≤ h
We use the representation of dˆN −1 and "interchange" again the order of the supremum and the
integral. Hence
Z
IT
γ
γ
ˆ
sup
hc̃ + (1 − hc̃)
dN −1 (iT , y)Q̂N −1 (diT | y)
1
0≤c̃≤ h
1
0≤c̃≤ h
(
× sup
"
≥ sup
α∈D
hc̃γ + (1 − hc̃)γ erhγ
⊤
(1 + α (e
−rh
γ
R(z) − 1d ))
Q̂Z
N −1 (dz
| iT , y)Q̂INT−1 (diT
)
| y)
=da=0
N −1 (iT , y).
For the last equality we use the tower property of the conditional expectation.
R
T
Inductive step: Let dn+1 (iT , y)Q̂In+1
(diT | y) ≥ da=0
n+1 (y) for all y ∈ Z (induction hypothesis).
n + 1 → n : Let y ∈ Z. By Theorem 4.5.3 we can write dn as follows
Z
dn (iT , y)Q̂InT (diT | y)
(
Z
=
sup
hc̃γ + (1 − hc̃)γ erhγ
1]
c̃∈[0, h
× sup
α∈D
Z
⊤
(1 + α (e
−rh
γ
R(z) − 1d ))
dn+1 (iT , y + z)Q̂Z
n (dz
)
| iT , y) Q̂InT (diT | y).
75
4 Discrete-Time Consumption-Investment Problems
By "interchanging" the order of the supremum and the expectation we obtain the following inequality
Z
dn (iT , y)Q̂InT (diT | y)
(
hc̃γ + (1 − hc̃)γ erhγ
≥ sup
1]
c̃∈[0, h
"
× sup
α∈D
1]
c̃∈[0, h
(
× sup
"
= sup
α∈D
⊤
(1 + α (e
−rh
γ
dn+1 (iT , y + z)Q̂Z
n (dz
γ
T
dn+1 (iT , y + z)Q̂In+1
(diT
R(z) − 1d ))
| iT , y)Q̂InT (diT
)
| y)
hc̃γ + (1 − hc̃)γ erhγ
⊤
(1 + α (e
−rh
R(z) − 1d ))
|
y + z)Q̂Z
n (dz
)
| y) ,
where we use Lemma 4.6.6 for the last equality. Eventually, we obtain the assertion by the
induction hypothesis
(
hc̃γ + (1 − hc̃)γ erhγ
sup
1]
c̃∈[0, h
× sup
α∈D
≥ sup
1]
c̃∈[0, h
× sup
α∈D
Z
⊤
(1 + α (e
−rh
γ
R(z) − 1d ))
(
Z
T
(diT
dn+1 (iT , y + z)Q̂In+1
| y + z)Q̂Z
n (dz
)
| y)
hc̃γ + (1 − hc̃)γ erhγ
Z
)
Z
(1 + α⊤ (e−rh R(z) − 1d ))γ da=0
n+1 (y + z)Q̂n (dz | y)
=da=0
n (y).
Consequently, for n = 0 it holds
Z
(0)
d0 (iT , 0)QI0T (diT ) ≥ da=0
0
and thus we obtain the second statement by Theorem 4.6.11.
Analogously to Theorem 4.6.9 it follows:
Theorem 4.6.13. Let Σ0 and Σε be simultaneously diagonalizable. Moreover, let a′ ∈ [0, 1) be
Y
the convex factor of ITX and let a ∈ [0, 1) be the convex factor of ITY . If ΣX
a′ = Σa , then the investor
X
Y
is indifferent between the insider information IT and IT (i.e. CEXY = 0).
Certainty Equivalent in the Numerical Example (Power Utility)
Figure 4.7
In Figure 4.7 we plot the certainty equivalent CEIT and CEXY against the public uncertainty
Σ0 . We fix λ0 = 0.3 and consider the risk aversion coefficients γ = 0.6 and γ = 0.01. All the
76
4.6 Properties of the Portfolio Value and the Optimal Strategy
γ = 0.01
0.06
0.06
0.04
0.04
0.02
CES
CEW
CEλ
0
−0.02
−0.04
0
0.5
γ = 0.6
1
1.5
0.02
−0.02
−0.04
2
0.04
0.04
0.02
0
−0.02
−0.02
1
1.5
2
1
1.5
2
1.5
2
Σ0
0.02
0
0.5
0.5
γ = 0.6
0.06
0
0
Σ0
0.06
−0.04
CEW S
CEλS
CEλW
0
CEXY
CEIT
CEXY
CEIT
γ = 0.01
−0.04
0
0.5
Σ0
1
Σ0
Figure 4.7: Certainty equivalent in dependence on Σ0 for different γ
remaining parameters are chosen as in Figure 4.4 and we observe here also a similar behavior.
However, especially for stock price information, it seems that the investor with power utility
functions with γ = 0.6 has always a higher certainty equivalent compared to an investor with
logarithmic utility functions (for the same values of λ0 and Σ0 ). The risk aversion coefficient γ
indicates the risk preferences of the investor: The higher γ, the less risk averse is the investor
(compare Arrow-Pratt measure). An investor with power utility functions is more risk loving than
an investor with logarithmic utility functions. Hence, it is intuitively plausible that the investor
with power utility functions is able to derive a higher value out of the additional information. The
difference between the two certainty equivalents (CEIT (power) − CEIT (log)) can be interpreted
as a "compensation" for taking on a more risky strategy.
Optimal Strategy in the Numerical Example (Power Utility)
Figure 4.8 and Figure 4.9
In Figure 4.8 we plot the average investment strategy against the risk aversion coefficient γ for
different values of Σ0 and λ0 . The same plot contains the optimal investment strategy of a regular
investor. The investor’s risk aversion is decreasing in γ. We observe that the less risk averse the
investor is, the more he invests into the stock whether or not he has additional information. The
strategy without insider information seems to have a kink: The more risk loving the investor, the
more willing he is to take on a more risky strategy by investing more into the risky asset, however,
since short selling is not optimal it comes to the observed kink. In the first row of the figure we
77
4 Discrete-Time Consumption-Investment Problems
keep λ0 at 0.1. If the risk aversion coefficient is small enough (γ < 0.4), then the insider takes
a more aggressive stock position relatively to the regular investor. This effect is more enhanced
when the investor has information about the stock price rather than about the Brownian motion.
This seems to be reasonable since contrary to Brownian, stock price information contain beside
the future information implicitly information about the unknown drift. However, when the risk
aversion becomes smaller (γ increases) we observe a quite unexpected effect. An investor with
stock price information takes a less aggressive stock position as a regular investor and as an
investor with Brownian information. This effect seems even more marked when we enhance the
public uncertainty to 1.
Σ0 = 1, λ0 = 0.1
1
0.8
0.8
0.6
0.6
ᾱ0∗
ᾱ0∗
Σ0 = 0.5, λ0 = 0.1
1
0.4
0.2
0.4
0.2
0
0
0
1
0.5
0
γ
Σ0 = 0.5, λ0 = 0.3
1
Σ0 = 1, λ0 = 0.3
1
1
0.8
0.8
0.6
0.6
Stock
0.4
ᾱ0∗
ᾱ0∗
0.5
γ
Brownian
Drift
0.2
0.4
0.2
without
0
0
0
1
0.5
0
γ
0.5
1
γ
Figure 4.8: Average optimal strategy in dependence on γ for different Σ0 and λ0
In Figure 4.9 we plot the average optimal strategy against the public uncertainty Σ0 for different
values of the convex factor and different values of the risk aversion coefficient. Due to the relation
xγ − 1
= log(x)
γ→0
γ
lim
it seems to be natural that we obtain for γ = 0.01 similar results as for logarithmic utility functions
(compare Figure 4.5). If we increase the risk aversion coefficient γ (risk aversion decreases), then
the differences in the optimal strategies seem to be more significant.
78
4.6 Properties of the Portfolio Value and the Optimal Strategy
a = 0.75, γ = 0.01
1
0.5
0.5
ᾱ0∗
ᾱ0∗
a = 0.25, γ = 0.01
1
0
0
2
1
0
2
Σ0
a = 0.75, γ = 0.6
1
1
0.5
Stock
Brownian
Drift
without
ᾱ0∗
ᾱ0∗
1
0
Σ0
a = 0.25, γ = 0.6
0
0.5
0
2
1
0
1
0
Σ0
2
Σ0
Figure 4.9: Average optimal strategy in dependence on Σ0 for different a and γ
4.6.2 Monotonicity Results for d = 1
In this section we look at the sensitivity of the optimal values and the optimal consumptioninvestment strategies towards changes in the realized value of the insider information and the
observation. Here, we concentrate on the special case d = 1, i.e. the investor can only invest in a
bond and in one stock driven by a one-dimensional Brownian motion. We obtain our monotonicity
results by using ideas and results from Topkis’ monotonicity theorem (see for instance Topkis
[1978]) and from Proposition 2.4.16 in Bäuerle & Rieder [2011].
The portfolio values and the strategies depend through the conditional expectation of the observation process Z on the insider information iT and the observation y (see Table 4.3). Therefore,
we first look at the behavior of Q̂Z
n (· | iT , y) towards changes in iT and y. For this purpose, we introduce some stochastic orders as studied in Müller & Stoyan [2002] and provide some properties
formulated in Lemma 4.6.15.
d
d
Definition 4.6.14. Let X = PX , Y = PY and let fX , fY be their respective densities with respect
to the Lebesgue measure. Then
a) X is said to be smaller than Y with respect to the (usual) stochastic order (written
PX ≤st PY ) if for all t ∈ R
PX ((−∞, t]) ≥ PY ((−∞, t]);
79
4 Discrete-Time Consumption-Investment Problems
b) X is said to be smaller than Y with respect to the likelihood ratio order (written PX ≤lr
PY ) if for all t ≤ t′ ∈ R
fX (t′ )fY (t) ≤ fX (t)fY (t′ ).
c) X is said to be smaller than Y in the increasing convex order (written PX ≤icx PY )
if for all increasing and convex functions f (for which the following expectations exist) it
holds
E[f (X)] ≤ E[f (Y )].
For a proof of the following properties we refer to Müller & Stoyan [2002] Theorem 1.3.8 and
Theorem 1.4.4.
d
d
Lemma 4.6.15. Let X = PX and Y = PY .
a) If PX ≤lr PY , then PX ≤st PY ;
b) The following statements are equivalent:
(i) PX ≤st PY ;
(ii) For all increasing functions f (for which the following expectations exist) it holds
E[f (X)] ≤ E[f (Y )].
d
d
2 ) and Y = N (µ , σ 2 ). Then P ≤
c) Let X = N (µX , σX
icx PY if and only if µX ≤ µY and
Y
X
Y
2
2
σX ≤ σY .
The following lemma can be interpreted as a higher realized value of the insider information makes
a higher future observation and thus a higher future stock price more likely.
Z
′
′
Lemma 4.6.16. Q̂Z
n (· | iT , y) ≤lr Q̂n (· | iT , y) if and only if iT ≤ iT .
Proof. From Definition 4.6.14 it follows that
Z
′
Q̂Z
n (· | iT , y) ≤lr Q̂n (· | iT , y)
⇔
q̂nZ (z | i′T , y)
is increasing in z.
q̂nZ (z | iT , y)
Therefore, we consider
q̂nZ (z | i′T , y)
q̂nZ (z | iT , y)
= exp
Z
q̂n
(z|i′T ,y)
Z
q̂n (z|iT ,y)
C IT (i′T − iT )
(1 + (n + 1)hΣIaT )
z−
0.5(C IT )2 h((i′T )2 − i2T ) + (i′T − iT )(CλIT λ0 + ΣIaT y)C IT h
(1 + nhΣIaT )(1 + (n + 1)hΣIaT )
is increasing in z if and only if
C IT (i′T −iT )
I
!
.
≥ 0. Since (1 + (n + 1)hΣIaT ) > 0 for
(1+(n+1)hΣaT )
and C IT > 0 for a
n = 0, . . . , N − 1 by Assumption 4.3.24
> 0 (C IT = 0 for a = 0 but in this
Z
′
case there is no insider information) it follows that Q̂n (· | iT , y) ≤lr Q̂Z
n (· | iT , y) if and only if
′
iT ≤ iT .
IT
Z
′
′
Lemma 4.6.17. Q̂Z
n (· | iT , y) ≤lr Q̂n (· | iT , y ) if and only if Σa (y − y) ≥ 0.
80
4.6 Properties of the Portfolio Value and the Optimal Strategy
Proof. The assertion follows since
Z
q̂n
(z|iT ,y ′ )
Z
q̂n (z|iT ,y)
is increasing in z if and only if ΣIaT (y ′ − y) ≥ 0.
A higher sum of past observations y comes along with higher past stock prices. Therefore, we
intuitively interpret the statement of Lemma 4.6.17 as follows: If ΣIaT ≥ 0, then a higher y at
time tn = nh makes a higher (future) stock price at time tn+1 = (n + 1)h more likely, however, if
ΣIaT ≤ 0, then a smaller y at time tn makes a higher stock price at time tn+1 more likely.
IT
Z
′
′
IT ′
′
Lemma 4.6.18. Q̂Z
n (· | iT , y) ≤lr Q̂n (· | iT , y ) if and only if C (iT − iT ) + Σa (y − y) ≥ 0.
Proof. The assertion follows since
ΣIaT (y ′ − y) ≥ 0.
Z
q̂n
(z|i′T ,y ′ )
Z
q̂n (z|iT ,y)
is increasing in z if and only if C IT (i′T − iT ) +
Z
′
Z
Z
′
Corollary 4.6.19. Let Q̂Z
n (· | iT , y) ≤lr Q̂n (· | iT , y) and Q̂n (· | iT , y) ≤lr Q̂n (· | iT , y ). Then
Z
′ ′
Q̂Z
n (· | iT , y) ≤lr Q̂n (· | i , y ).
The reverse statement of the Corollary is in general not true.
Proof. Since C IT ≥ 0 the statement follows from Lemma 4.6.16, Lemma 4.6.17 and Lemma 4.6.18.
IT
IT
IT
IT
Lemma 4.6.20. Q̂Z
n (· | iT , y) ≥icx Q̂n+1 (· | iT , y) if and only if Σa (Cλ + C iT + Σa y) ≥ 0.
Z
Proof. For the expectation of Q̂Z
n and Q̂n+1 it holds
h
CλIT + C IT iT + ΣIaT y
1 + nhΣIaT
≥h
CλIT + C IT iT + ΣIaT y
1 + (n + 1)hΣIaT
if and only if ΣIaT (CλIT + C IT iT + ΣIaT y) ≥ 0. For the variance it always holds
h
1 + (n + 1)hΣIaT
1 + nhΣIaT
≥h
The assertion follows then by Lemma 4.6.15.
1 + ((n + 1) + 1)hΣIaT
1 + (n + 1)hΣIaT
.
Logarithmic Utility Functions
The behavior of the value function with respect to changes in the realized value of the insider
information depends on ΣIaT . Whereas for drift information the modified covariance matrix is
non-negative, for Brownian and stock price information ΣaIT can also be negative.
Theorem 4.6.21. Let iT ≤ i′T , y ≤ y ′ and let ΣIaT ≥ 0. Then dn (iT , y) ≤ dn (i′T , y ′ ) and therefore
Vn (x, iT , y) ≤ Vn (x, i′T , y ′ ) for all x > 0.
Proof. We show the statement by a backward induction. Let (iT , y) ∈ I × Z with iT ≤ i′T and
y ≤ y′ .
Initial step:
n = N : X dN (iT , y) = 0 = dN (i′T , y ′ );
n = N − 1 : X since:
81
4 Discrete-Time Consumption-Investment Problems
σ2
As σ > 0 the function g(z) := log(1 + α(eσz−(r+ 2 )h − 1)) is for all α ∈ D monotonically increasing.
Thus, by Lemma 4.6.18 and Lemma 4.6.15 we obtain the following inequality
Z
σ2
∗
dN −1 (iT , y) =rh + sup CN (c̃) + sup log(1 + α(eσz−(r+ 2 )h − 1))Q̂Z
N −1 (dz | iT , y)
1
0<c̃< h
α∈D
∗
≤rh + sup CN
(c̃) + sup
1
0<c̃< h
α∈D
Z
log(1 + α(eσz−(r+
σ 2 )h
2
′
′
− 1))Q̂Z
N −1 (dz | iT , y ).
Inductive step: Let dn+1 (iT , y) ≤ dn+1 (i′T , y ′ ) for all i, i′ ∈ I with i ≤ i′ and y, y ′ ∈ Z with y ≤ y ′
(induction hypothesis).
n + 1 → n : We obtain the following first inequality using the same arguments as in the initial
step. Then for the second inequality we use the induction hypothesis
Z
∗
(c̃) + dn+1 (iT , y + z)Q̂Z
dn (iT , y) =((N − (n + 1))h + 1)rh + sup Cn+1
n (dz | iT , y)
1
0<c̃< h
Z
σ2
log(1 + α(eσz−(r+ 2 )h − 1))Q̂Z
n (dz | iT , y)
α∈D
Z
∗
(c̃) + dn+1 (iT , y + z)Q̂Z
≤((N − (n + 1))h + 1)rh + sup Cn+1
n (dz | iT , y)
+ ((N − (n + 1))h + 1) sup
1
0<c̃< h
Z
σ2
′ ′
log(1 + α(eσz−(r+ 2 )h − 1))Q̂Z
n (dz | i , y )
α∈D
Z
∗
≤((N − (n + 1))h + 1)rh + sup Cn+1 (c̃) + dn+1 (i′T , y ′ + z)Q̂Z
n (dz | iT , y)
+ ((N − (n + 1))h + 1) sup
1
0<c̃< h
+ ((N − (n + 1))h + 1) sup
α∈D
Z
log(1 + α(eσz−(r+
σ 2 )h
2
′ ′
− 1))Q̂Z
n (dz | i , y ).
It follows from the induction hypothesis that dn (i′T , y ′ + z) is monotonically increasing in z. An
application of Lemma 4.6.18 and Lemma 4.6.15 gives us the assertion
Z
∗
((N − (n + 1))h + 1)rh + sup Cn+1
(c̃) + dn+1 (i′T , y ′ + z)Q̂Z
n (dz | iT , y)
1
0<c̃< h
+ ((N − (n + 1))h + 1) sup
α∈D
Z
log(1 + α(eσz−(r+
σ 2 )h
2
′ ′
− 1))Q̂Z
n (dz | i , y )
∗
≤((N − (n + 1))h + 1)rh + sup Cn+1
(c̃)
1
0<c̃< h
Z
′
′
dn+1 (i′T , y ′ + z)Q̂Z
n (dz | iT , y )
Z
σ2
′
′
+ ((N − (n + 1))h + 1) sup log(1 + α(eσz−(r+ 2 )h − 1))Q̂Z
n (dz | iT , y )
+
α∈D
=dn (i′T , y ′ ).
It results from Theorem 4.6.21 that for ΣIaT ≥ 0 the value of the h-insider’s consumption-investment
problem is increasing in the realized value of the insider information
Corollary 4.6.22. Let iT ≤ i′T and let ΣIaT ≥ 0. Then dn (iT , y) ≤ dn (i′T , y) and therefore
V0 (x0 , iT , 0) ≤ V0 (x0 , i′T , 0).
82
4.6 Properties of the Portfolio Value and the Optimal Strategy
Remark 4.6.23. Note that for ΣIaT ≤ 0 it is in general not true that dn (iT , y) ≥ dn (i′T , y ′ ) for all
iT ≤ i′T and y ≤ y ′ . For this purpose, we consider dN −1 (iT , y). Let ΣIaT ≤ 0, iT ≤ i′T and y ≤ y ′ .
• If C IT (i′T − iT ) + ΣIaT (y ′ − y) ≥ 0, then dN −1 (iT , y) ≤ dN −1 (i′T , y ′ );
{z
}
|
{z
} |
≤0
≥0
• If C IT (i′T − iT ) + ΣIaT (y ′ − y) ≤ 0, then dN −1 (iT , y) ≥ dN −1 (i′T , y ′ ).
However, for all ΣIaT ∈ R and for iT ≤ i′T we have dN −1 (iT , y) ≤ dN −1 (i′T , y). Hence, if N = 1,
then V0 (x0 , iT , 0) ≤ V0 (x0 , i′T , 0).
∗)
We consider the optimal consumption-investment strategy π ∗ = (c∗n , αn
n∈{0,...,N } . The optimal
consumed fraction of wealth is a deterministic function which is increasing in time but does not
change in the realized value of the insider information and the observation. Therefore, we only
∗)
consider the behavior of the optimal investment strategy α∗ = (αn
n∈{0,...,N −1} with respect to
changes in iT and y. The following theorem shows: If ΣIaT ≥ 0, then the optimal investment
strategy is increasing in (iT , y) (on the partially ordered set I × Z where (iT , y) ≤ (i′T , y ′ ) :⇔
iT ≤ i′T and y ≤ y ′ ). However, due to the short selling constraints the strategy is not strictly
monotone.
Theorem 4.6.24. Consider the optimal investment strategy as a function of the insider information and the observation (iT , y) at a fixed time nh
Z
σ2
∗
log(1 + α(eσz−(r+ 2 )h − 1))Q̂Z
αn
(iT , y) = arg max
n (dz | iT , y).
α∈[0,1]
Then it holds:
∗ (i , y) ≤ α∗ (i′ , y ′ )
a) αn
T
n T
if
C IT (i′T − iT ) + ΣIaT (y ′ − y) ≥ 0;
∗ (i , y) ≥ α∗ (i′ , y ′ )
b) αn
T
n T
if
C IT (i′T − iT ) + ΣIaT (y ′ − y) ≤ 0.
As aforementioned, a higher value of y comes along with higher values of past stock prices and
a higher value of iT lets us suppose a higher terminal stock price. Moreover, in view of Lemma
4.6.18, if ΣIaT ≥ 0, then higher values of iT and y make a higher future stock price more likely.
Hence, it seems to be intuitive: the higher the values of iT and y, the higher is the fraction of
wealth invested into the stock (C IT > 0).
Proof. Let α ≤ α′ ∈ D = [0, 1]. We consider the function
g(z) := log(1 + α′ (eσz−(r+
σ 2 )h
2
− 1)) − log(1 + α(eσz−(r+
σ 2 )h
2
− 1)).
(4.38)
The latter is monotonically increasing since
′
g (z) =
1 + α(eσz−(r+
1 + α′ (e
σ 2 )h
2
2
σz−(r+ σ2 )h

×
− 1)
− 1)
1 + α(eσz−(r+
with g ′ (z) ≥ 0 if and only if α′ ≥ α.
2
2
σ
α′ σeσz−(r+ 2 )h
σ 2 )h
2
− 1)
−
2
σ
σ
(1 + α′ (eσz−(r+ 2 )h − 1))ασeσz−(r+ 2 )h
(1 + α(eσz−(r+
σ 2 )h
2
− 1))2


83
4 Discrete-Time Consumption-Investment Problems
a) Let iT , i′T ∈ I and y, y ′ ∈ Z with C IT (i′T − iT ) + ΣIaT (y ′ − y) ≥ 0. Then it follows from
Z
′
′
Lemma 4.6.18 that Q̂Z
n (· | iT , y) ≤lr Q̂n (· | iT , y ). Since the likelihood ratio order implies
the stochastic order and g is increasing, it follows from Lemma 4.6.15 that
Z
Z
Z
′
′
g(z)Q̂n (dz | iT , y) ≤ g(z)Q̂Z
(4.39)
n (dz | iT , y ).
Moreover, we define
hn (iT , y, α) :=
Z
log(1 + α(eσz−(r+
σ 2 )h
2
− 1))Q̂Z
n (dz | iT , y).
Then it follows from (4.39) that
hn (iT , y, α′ ) − hn (iT , y, α) ≤ hn (i′T , y ′ , α′ ) − hn (i′T , y ′ , α)
(4.40)
∗ (i , y) of h (i , y, ·)
for α ≤ α′ . As shown in the proof of Theorem 4.5.1 the maximizer αn
n T
T
∗
∗ (i′ , y ′ ) and (ii)
is uniquely determined. We consider the two possibilities (i) αn (iT , y) ≤ αn
T
∗ (i , y) ≥ α∗ (i′ , y ′ ) separately:
αn
T
n T
∗ (i , y) ≤ α∗ (i′ , y ′ ). Then
(i) Let αn
T
n T
∗
∗ ′
∗ ′
max{αn
(iT , y), αn
(iT , y ′ )} = αn
(iT , y ′ ) and
∗
∗ ′
′
∗ ′
hn (iT , y, max{αn (iT , y), αn (iT , y )}) − hn (iT , y, αn
(iT , y ′ )) = 0;
∗ (i , y) ≥ α∗ (i′ , y ′ ). Then
(ii) Let αn
T
n T
∗
∗ ′
∗
max{αn
(iT , y), αn
(iT , y ′ )} = αn
(iT , y) and
∗
∗ ′
′
∗ ′
hn (iT , y, max{αn (iT , y), αn (iT , y )}) − hn (iT , y, αn
(iT , y ′ )) ≥ 0
∗ (i , y) is the maximizer of h (i , y, ·).
since αn
n T
T
By (i) and (ii) we obtain
∗
∗ ′
∗ ′
hn (iT , y, max{αn
(iT , y), αn
(iT , y ′ )}) − hn (iT , y, αn
(iT , y ′ )) ≥ 0.
We conclude by (4.40)
∗ ′
∗
∗ ′
hn (i′T , y ′ , max{αn
(iT , y ′ )}) − hn (i′T , y ′ , αn
(iT , y), αn
(iT , y ′ ))
∗
∗ ′
∗ ′
≥hn (iT , y, max{αn
(iT , y), αn
(iT , y ′ )}) − hn (iT , y, αn
(iT , y ′ ))
≥0.
∗ (i , y), α∗ (i′ , y ′ )} maximizes h (i′ , y ′ , ·). But this maximizer is unique
Therefore, max{αn
n T
T
n T
∗
∗ (i′ , y ′ ) = max{α∗ (i , y), α∗ (i′ , y ′ )} and α∗ (i′ , y ′ ) ≥
and given by αn (i′T , y ′ ). Hence, αn
n T
n T
n T
T
∗ (i , y).
αn
T
b) Let iT , i′T ∈ I and y, y ′ ∈ Z with C IT (i′T − iT ) + ΣIaT (y ′ − y) ≤ 0. Then by Lemma 4.6.18 it
Z
′
′
holds that Q̂Z
n (· | iT , y) ≥lr Q̂n (· | iT , y ). Since g is increasing it follows from Lemma 4.6.15
that
Z
Z
′
′
g(z)Q̂Z
(dz
|
i
,
y)
≥
g(z)Q̂Z
T
n
n (dz | iT , y ).
The rest of the proof follows the same lines as part a).
84
4.6 Properties of the Portfolio Value and the Optimal Strategy
Now let us consider the observation as fixed. Then the optimal investment strategy is increasing
in the realized value of the insider information, i.e. the higher the value of the insider information,
the more the h-insider invests into the stock.
Corollary 4.6.25. Consider the optimal investment strategy as a function of the insider information iT at a fixed time nh and for a fixed observation y. Then for iT ≤ i′T it holds
∗
∗ ′
0 ≤ αn
(iT , y) ≤ αn
(iT , y) ≤ 1.
Proof. This Corollary follows directly from Theorem 4.6.24 for y = y ′ . Note that in this case (4.40)
is given by
hn (iT , y, α′ ) − hn (iT , y, α) ≤ hn (i′T , y, α′ ) − hn (i′T , y, α),
i.e. the differences of h in α are increasing in iT . The function h is then said to have increasing
differences.
The optimal investment strategy of a regular investor is always increasing in the observation y.
However, as it follows from Theorem 4.6.24 (with iT = i′T ), this is in general not true for the
insider. The behavior of the optimal strategy with respect to changes in y depends on the sign of
ΣIaT . If ΣIaT ≥ 0, then for a given value of the insider information the h-insider invests the more
into the stock the higher y. On the other hand, if ΣIaT ≤ 0, then the h-insider invests the more
into the stock the smaller the value of the observation.
Corollary 4.6.26. Consider the optimal investment strategy as a function of the observation y
at a fixed time nh and for a fixed value of the insider information iT . Then for y ≤ y ′ it holds:
∗ (i , y) ≤ α∗ (i , y ′ )
(a) αn
T
n T
∗ (i , y) ≥ α∗ (i , y ′ )
(b) αn
T
n T
if
if
ΣIaT ≥ 0;
ΣIaT ≤ 0.
Power Utility Functions
We consider an h-insider with power utility functions. We obtain similar results as for logarithmic
utility functions.
Theorem 4.6.27. Let iT ≤ i′T , y ≤ y ′ and let ΣIaT ≥ 0. Then dn (iT , y) ≤ dn (i′T , y ′ ) and therefore
Vn (x, iT , y) ≤ Vn (x, i′T , y ′ ) for all x ≥ 0.
Remark 4.6.28. For ΣIaT ≤ 0 compare Remark 4.6.23.
Proof. We show the statement by a backward induction. Let (iT , y) ∈ I × Z with iT ≤ i′T and
y ≤ y′
Initial step:
n = N : X dN (iT , y) = 1 = dN (i′T , y ′ );
n = N − 1 : X since:
σ2
As σ > 0 the function g(z) := (1 + α(eσz−(r+ 2 )h − 1))γ is for all α ∈ D monotonically increasing
and non-negative. Hence, by Lemma 4.6.18 together with Lemma 4.6.15 we obtain the following
inequality
!1−γ
Z
dN −1 (iT , y) = h + erhγ sup
α∈D
(1 + α(eσz−(r+
σ 2 )h
2
δ
− 1))γ Q̂Z
N −1 (dz | iT , y)
δ !1−γ
Z
2
γ Z
′
′
rhγ
σz−(r+ σ2 )h
− 1)) Q̂N −1 (dz | iT , y )
.
≤ h+ e
sup (1 + α(e
α∈D
85
4 Discrete-Time Consumption-Investment Problems
Inductive step: Let dn+1 (iT , y) ≤ dn+1 (i′T , y ′ ) for all iT , i′T ∈ I with iT ≤ i′T and y, y ′ ∈ Z with
y ≤ y ′ (induction hypothesis).
n + 1 → n : By the induction hypothesis we have dn+1 (iT , y + z) ≤ dn+1 (i′T , y ′ + z). Hence
δ !1−γ
Z
2
γ
Z
σz−(r+ σ2 )h
rhγ
− 1)) dn+1 (iT , y + z)Q̂n (dz | iT , y)
dn (iT , y) = h + e
sup (1 + α(e
α∈D
δ !1−γ
Z
2
rhγ
σz−(r+ σ2 )h
γ
′
′
Z
≤ h+ e
sup (1 + α(e
− 1)) dn+1 (iT , y + z)Q̂n (dz | iT , y)
.
α∈D
Moreover, it follows from the induction hypothesis that dn+1 (i′T , y ′ + z) ≤ dn+1 (i′T , y ′ + z ′ ) for
z ≤ z ′ . Since dn+1 and g are non-negative, the function g(z)dn+1 (i′T , y ′ + z) is increasing in z as
the product of non-negative increasing functions. Applying Lemma 4.6.18 together with Lemma
4.6.15 gives us the assertion
δ !1−γ
Z
2
γ
′
′
Z
rhγ
σz−(r+ σ2 )h
− 1)) dn+1 (iT , y + z)Q̂n (dz | iT , y)
h+ e
sup (1 + α(e
α∈D
δ !1−γ
Z
2
γ
′
′
Z
′
′
σz−(r+ σ2 )h
rhγ
− 1)) dn+1 (iT , y + z)Q̂n (dz | iT , y )
≤ h+ e
sup (1 + α(e
α∈D
=dn (i′T , y ′ ).
From Theorem 4.6.27 we immediately obtain:
Corollary 4.6.29. Let iT ≤ i′T and let ΣIaT ≥ 0. Then dn (iT , y) ≤ dn (i′T , y) and therefore
V0 (x0 , iT , 0) ≤ V0 (x0 , i′T , 0).
∗)
We consider the optimal consumption-investment strategy π = (c∗n , αn
n∈{0,...,N −1} for an hinsider with power utility functions. The optimal consumed fraction of wealth is independent of
the wealth. Hence, we look at the optimal consumed fraction of wealth c̃n only with respect to
changes in (iT , y).
Theorem 4.6.30. Consider the optimal consumed fraction of wealth as a function of (iT , y) at
a fixed time nh, i.e.
c̃n (iT , y) =
1
(dn (iT , y))δ
.
Let ΣIaT ≥ 0. Then for iT ≤ i′T and y ≤ y ′ it holds
c̃n (iT , y) ≥ c̃n (i′T , y ′ ).
Proof. The statement follows from Theorem 4.6.27.
If ΣIaT ≥ 0, then higher values of iT and y make a higher future stock price more likely. Hence, it
seems to be intuitive that the optimal consumption is then decreasing in (iT , y) (on the partially
ordered set I × Z) since the less the investor consumes the more can be invested into the financial
market. In a financial market without parameter uncertainty and without insider information
86
4.6 Properties of the Portfolio Value and the Optimal Strategy
it can easily be seen that dn is decreasing in n (set a = 0 and Σ0 = 0 in order to see that Q̂Z
n
is independent of n), i.e. the closer the investment horizon comes the higher is the consumed
fraction of wealth. In the setting with partial and insider information the investor also increases
the consumed fraction of wealth (under some assumptions) immediately before his investment
horizon is reached:
Theorem 4.6.31. Let (CλIT λ0 + C IT iT + ΣIaT y)ΣIaT ≥ 0. Then c̃N −2 (iT , y) ≤ c̃N −1 (iT , y).
Proof. Let f (α, z) :=
sup
Z
α∈D
γ
2
σ
1 + α(eσz−(r+ 2 )h − 1)
. Then
f (α, z)Q̂Z
N −1 (dz | iT , y) ≥
Z
f (0, z)Q̂Z
N −1 (dz | iT , y) = 1.
Hence, for all iT , y it holds
dN −1 (iT , y) = h + e
rhγ
sup
α∈D
Z 1 + α(e
2
σz−(r+ σ2 )h
γ
δ !1−γ
Z
− 1) Q̂N −1 (dz | iT , y)
≥1 = dN (iT , y)
and consequently
dN −2 (iT , y) ≥
γ
δ !1−γ
Z 2
σ
.
1 + α(eσz−(r+ 2 )h − 1) Q̂Z
h + erhγ sup
N −2 (dz | iT , y)
α∈D
The function f (α, ·) is monotonically increasing and convex for all α ∈ D. Moreover, by assumption
we have (CλIT λ0 + C IT iT + ΣIaT y)ΣIaT ≥ 0. Therefore, by Lemma 4.6.20 together with Definition
R
R
4.6.14 it follows that f (α, z)Q̂N −2 (dz | iT , y) ≥ f (α, z)Q̂N −1 (dz | iT , y) and thus we obtain
γ
δ !1−γ
Z 2
Z
σz−(r+ σ2 )h
rhγ
− 1) Q̂N −2 (dz | iT , y)
1 + α(e
h+ e
sup
α∈D
γ
δ !1−γ
Z 2
Z
rhγ
σz−(r+ σ2 )h
− 1) Q̂N −1 (dz | iT , y)
≥ h+ e
sup
1 + α(e
α∈D
=dN −1 (iT , y).
The assertion follows due to c̃n (iT , y) = (dn (iT , y))−δ .
Theorem 4.6.32. Consider the optimal investment strategy as a function of (iT , y) at a fixed
time nh
Z γ
σ2
∗
1 + α(eσz−(r+ 2 )h − 1) dn+1 (iT , y + z)Q̂Z
αn
(iT , y) = arg max
(dz
|
i
,
y)
.
T
n
α∈D
Then it holds for the last investment decision at time t = (N − 1)h
∗
∗
′
′
a) αN
−1 (iT , y) ≤ αN −1 (iT , y )
if
C IT (i′T − iT ) + ΣIaT (y ′ − y) ≥ 0;
′
′
∗
∗
b) αN
−1 (iT , y) ≥ αN −1 (iT , y )
if
C IT (i′T − iT ) + ΣIaT (y ′ − y) ≤ 0.
87
4 Discrete-Time Consumption-Investment Problems
Proof. Note that dN (iT , y) = 1. We define for α ≤ α′ ∈ D
g(z) :=
(1 + α′ (eσz−(r+
σ 2 )h
2
− 1))γ − (1 + α(eσz−(r+
σ 2 )h
2
− 1))γ
with derivative
g ′ (z) =
γσα′ eσz−(r+
−γσαe
Since g̃(α) := α(1 + α(eσz−(r+
σ 2 )h
2
σ 2 )h
2
(1 + α′ (eσz−(r+
2
σz−(r+ σ2 )h
(1 + α(e
σ 2 )h
2
2
σz−(r+ σ2 )h
− 1))γ−1
− 1))γ−1 .
− 1))γ−1 is increasing on [0, 1] it follows that g ′ (z) ≥ 0.
a) Let iT , i′T ∈ I and y, y ′ ∈ Z with C IT (i′T − iT ) + ΣIaT (y ′ − y) ≥ 0. Then by Lemma 4.6.18
Z
Z
Z
′
′
g(z)Q̂N −1 (dz | iT , y) ≤ g(z)Q̂Z
N −1 (dz | iT , y ).
Now the assertion follows along the same lines as the proof of Theorem 4.6.24.
b) Let iT , i′T ∈ I and y, y ′ ∈ Z with C IT (i′T − iT ) + ΣIaT (y ′ − y) ≤ 0. Then by Lemma 4.6.18
Z
Z
′
′
Z
g(z)Q̂N −1 (dz | iT , y) ≥ g(z)Q̂Z
N −1 (dz | iT , y ).
We refer again to the proof of Theorem 4.6.24.
4.6.3 Summarizing Remarks
We showed how the discrete-time consumption-investment problem with partial and insider information can be solved. Using the Bellman equation we reduced the problem to a stepwise
optimization problem where it is enough to consider Markovian strategies. Nevertheless, we were
not able to present an optimal consumption-investment strategy in closed form.
For logarithmic utility functions the optimal consumed fraction of the wealth is only a deterministic time-dependent function and is without effect on the h-investor’s investment decisions.
Hence, we obtained in the terminal wealth and the consumption-investment problem the same
optimal investment strategy which depends on the insider information and the observation (compare Theorem 4.5.1 and Theorem 4.5.2). For power utility functions the optimal investment and
the optimal consumption depend on the insider information and the observation. Moreover, here
the optimal investment strategies are different for the problem with and without consumption
(compare Theorem 4.5.3 and Theorem 4.5.5).
By virtue of our theoretical results (Theorem 4.5.1 and Theorem 4.5.3) we are able to compute
an optimal consumption-investment strategy using numerical optimization procedures. Therefore,
we also provided a numerical example to study properties of the strategies, the value functions
and the indifference prices. We found, for instance, that the modified covariance matrix ΣIaT
plays a crucial role in our model. If the investor has drift information or if the investor does
not have additional information, then ΣIaT is positive definite for any convex factor a ∈ [0, 1), any
investment horizon T > 0 and for any positive definite matrices Σ0 , Σε . However, if the investor
has future information about the stock price or the Brownian motion, then ΣIaT can also be
negative definite. As seen in the one-dimensional case (d = 1) for logarithmic and power utility
functions the sign of ΣIaT determines whether or not the optimal strategy and the value function
are increasing in y and iT (see e.g. Corollary 4.6.25).
88
5 Continuous-Time
Consumption-Investment Problems
In this chapter we investigate the continuous-time financial market introduced in Chapter 2. Without loss of generality we suppose a positive definite and symmetric volatility matrix σ (compare
Chapter 3). Hence,
λ := σ −1 µ ∈ Rd ,
i.e.
µk =
d
X
σkj λj , k = 1, . . . , d,
j=1
is well-defined. We then consider λ as the unknown drift with initial distribution Q0 =
N (λ0 , Σ0 ). Now we assume that the investor is allowed to trade and to observe the financial
market at any time until the investment horizon is reached. For t ∈ [0, T ] the observation process
is given by
Yt := λt + Wt .
(5.1)
We refer to such an investor as a continuous-time investor.
5.1 Formulation of the Optimization Problem
Analogously to the previous chapter we formulate in this section the optimization problem which
we solve in Section 5.5 for an investor with logarithmic and power utility functions.
Consumption-Investment Strategy
At any time t ∈ [0, T ] the continuous-time investor decides about his consumption and about
his investment into the financial market consisting of a bond and d stocks. Therefore, we call
a (2 + d)-dimensional adapted (with respect to an appropriate filtration) stochastic process π =
(ct , bt , at )t∈[0,T ] a (continuous-time) consumption-trading strategy, where the quantities have
the following meaning
• ct ≥ 0 is the rate at which the investor consumes at time t;
• bt ∈ R is the number of bonds B in the portfolio at time t;
• akt ∈ R, k = 1, . . . , d, is the number of stocks S k in the portfolio at time t.
As in the discrete-time case we restrict ourselves to self-financing consumption-trading strategies.
We follow here the definition of Korn & Korn [2001]:
Definition 5.1.1. A (continuous-time) consumption-trading strategy π = (ct , bt , at )t∈[0,T ] is called
self-financing if the corresponding wealth process X = (Xt )t∈[0,T ] satisfies
Xt = x0 +
d Z
X
k=1 0
t
aks dSsk +
Zt
0
bs dBs −
Zt
cs ds.
0
89
5 Continuous-Time Consumption-Investment Problems
Let π = (ct , bt , at )t∈[0,T ] be a self-financing consumption-trading strategy with wealth process
Xt > 0. For t ∈ [0, T ] we define for k = 1, . . . , d
αtk :=
k
ak
t St
Xt .
Then αtk is the fraction of wealth which is invested into stock k at time t. Hence, we obtain the
fraction of wealth which is invested into the bond at time t as
(1 − αt⊤ 1d ) =
bt Bt
.
Xt
A self-financing strategy π is fully described by ct and αt . Thus, we call the (1 + d)-dimensional
process π = (ct , αt )t∈[0,T ] a (continuous-time) consumption-investment strategy.
Wealth Process
Using the self-financing condition we obtain for the wealth process
dXt =
d
X
k=1
at dStk + bt dBt − ct dt.
Writing the wealth process with the invested fractions we obtain
dXt =
d
X
Xt αtk
k=1
=Xt
d
X
k=1
αtk
d
X
dStk
dBt
+
X
(1
−
αtk )
− ct dt
t
k
Bt
St
k=1
d
X
j=1
σkj (λj dt + dWtj ) + (1 −
d
X
k=1
αtk )Xt rdt − ct dt.
This stochastic differential equation (SDE) is called wealth SDE. In order to emphasize that Xt
is the wealth obtained under strategy π we write Xtπ , i.e.
X0π
dXtπ
:=
=
x0 ;
(Xtπ αt⊤ (σλ − r1d ) + rXtπ − ct )dt + Xtπ αt⊤ σdWt ,
(5.2)
t ∈ [0, T ].
Remark 5.1.2. In the terminal wealth problem (without consumption) the wealth SDE reduces
to
dXtπ = Xtπ (αt⊤ (σλ − r1d ) + r)dt + Xtπ αt⊤ σdWt .
Optimization Problem
The continuous-time investor aims to maximize the expected utility of terminal wealth and intermediate consumption given the additional information IT = iT (at time t = 0) and the initial
wealth X0π = x0 . We evaluate the terminal wealth by the utility function Up and the intermediate consumption by the utility function Uc . We denote the continuous-time optimization
90
5.1 Formulation of the Optimization Problem
problem by (P c ) :
#
 "
RT


π

E Uc (ct )dt + Up (XT ) | IT = iT → max;



 0
(P c ) Xtπ ≥ 0, t ∈ [0, T ];



X0π = x0 ;




π = (ct , αt )t∈[0,T ] admissible consumption-investment strategy.
Enlarged Filtration and Admissible Consumption-Investment Strategy
Due to the partial information the investor does not have access to the background filtration F.
However, the investor has additional information. Therefore, before we define the set of admissible
consumption-investment strategies we introduce different filtrations:
(1) The enlarged background filtration: Firstly, we initially enlarge the background filtration in order to include the insider information IT ∈ {ITS , ITW , ITλ }. Roughly speaking, by an
initial enlargement of F by the F-measurable random variable IT , denoted by
Ft ∨ σ(IT ) := σ(Ft ∪ σ(IT )),
we mean that for all t ∈ [0, T ] the σ-field generated by IT is "added" to Ft . More formally,
the σ-field Ft ∨ σ(IT ) is defined as the smallest σ-field containing both: Ft and σ(IT ). In
order that the enlarged filtration satisfies the usual conditions, we redefine the enlarged
filtration in the following way
Gt :=
\
ε>0
(Ft+ε ∨ σ(IT )).
We refer to the enlarged background filtration G = (Gt )t∈[0,T ] also as the insider filtration.
Obviously, it holds Ft ⊆ Gt .
(2) Insider information filtration: The insider gets his information by observing the stock
prices and by his additional information. Hence, the insider information filtration FIT Y =
(FtIT Y )t∈[0,T ] is generated by the insider information IT and the observation process Y =
(Yt )t∈[0,T ] (augmented by P-null sets of F), i.e.
FtIT Y := σ(IT , Yu : 0 ≤ u ≤ t) ∨ N .
with P(N ) = 0. If the investor does not have any additional information, then the information
filtration FY = (FtY )t∈[0,T ] is generated by Y (augmented by P-null sets) with FtY ⊆ FtIT Y .
The investor can make his decisions only on the basis of his information, i.e. an admissible
strategy for the insider has to be adapted with respect to FIT Y . In contrast to the h-investor
a continuous-time investor can immediately react to changes in the asset prices and readjust
his portfolio. Therefore, by virtue of the continuous stock price processes (without jumps) the
continuous-time investor is able to keep his wealth non-negative all the time even when he short
sells the stock or the bond. Hence, for the continuous-time investor short selling and borrowing
might be optimal.
Definition 5.1.3. An admissible (self-financing) continuous-time consumption-investment strategy is an FIT Y -progressively measurable stochastic process π = (ct , αt )t∈[0,T ] with
91
5 Continuous-Time Consumption-Investment Problems
ct ∈ R≥0 and αt ∈ Rd which ensures a non-negative wealth process and satisfies the integrability
conditions
ZT
0
ct dt < ∞,
ZT
0
αt⊤ αt dt < ∞.
We denote by At the set of all admissible consumption-investment strategies when starting at time t.
Remark 5.1.4. The integrability conditions in Definition 5.1.3 ensure that the integrals in the
wealth SDE are well-defined.
Approach
It is important to mention here, that the set of admissible consumption-investment strategies of
an insider is greater than that of a regular investor. If we consider the wealth SDE (5.2) for an
admissible strategy π = (ct , αt )t∈[0,T ] , then we integrate the FIT Y -adapted investment strategy
(αt )t∈[0,T ] (in general not F-adapted) with respect to the F-Brownian motion W. However, in general the F-Brownian motion W is not a Brownian motion with respect to the information filtration
FIT Y . Moreover, the wealth SDE depends on λ, but an admissible consumption-investment strategy is not allowed to depend on the unknown drift. Hence, (P c ) is not a standard consumptioninvestment problem. Nevertheless, by following here a similar approach as for the discrete-time
consumption-investment problem (compare Chapter 4), consisting of an enlargement and a filtering step, we are able to reduce (P c ) to a standard consumption-investment problem:
(Step 1) Using results from the theory of initial enlargement of filtrations we incorporate the
insider information into the optimization problem. More precisely, we look for a semimartingale decomposition of W with respect to G in order to give the Brownian integral
in (5.2) a meaning. In (Step 1), the enlargement step, the drift is supposed to be known;
(Step 2) In the second step we incorporate the partial information into the optimization problem
obtained after (Step 1) by using results from the theory of stochastic filtering, i.e. we
estimate the unknown drift on the basis of the insider information and the observations
of the stock prices;
(Step 3) (Step 1) and (Step 2) result in a standard consumption-investment problem with complete information. Due to the Markovian structure of the price processes we solve (P c )
for an investor with logarithmic and power utility functions via the Hamilton-JacobiBellman equation (HJB).
5.2 Enlargement and Filtering
In this section we reduce the optimization problem (P c ) with partial and insider information to a
standard consumption-investment problem with complete information. Throughout this chapter
we make use of the abbreviation
Kt := a2 (T − t)Id + (1 − a)2 Σε ,
t ∈ [0, T ].
Due to the positive definiteness of Σε the matrix-valued function Kt is for all a ∈ [0, 1) positive
definite.
92
5.2 Enlargement and Filtering
5.2.1 Enlargement
Comparable to the discrete-time consumption-investment problem we firstly incorporate only the
insider information and take λ as given, i.e. we consider the price processes under the enlarged
background filtration. As aforementioned the Brownian motion W is in general not a Brownian
motion with respect to G. However, we show in this section that there exists a G-adapted stochastic
process η IT = (ηtIT )t∈[0,T ] such that W IT = (WtIT )t∈[0,T ] with
WtIT
:= Wt −
Zt
ηsIT ds
(5.3)
0
is a Brownian motion with respect to the insider filtration G. The process η IT is the so-called
information drift. Examples of such information drifts, even for a more general setting, can be
found in Yor [1992].
Remark 5.2.1. The processes W IT and η IT are d-dimensional processes and the integral in
equation (5.3) is defined componentwise.
The information drift depends on the three different types of insider information. However, if it
is clear from the context which type of insider information we consider or if a result holds for
all three types of insider information, then we simply write η IT and W IT . Otherwise, we write
η S , η W , η λ and W S , W W , W λ for the respective type of information.
(I) Stock Price Information
We start with stock price information, i.e. IT = ITS . Therefore, the enlarged background filtration
G = (Gt )t∈[0,T ] is the one containing the stock price information, i.e.
Gt = Ft ∨ σ(ITS ).
Theorem 5.2.2. Let the G-adapted process η S = (ηtS )t∈[0,T ] be given by
ηtS := Kt−1 (aITS − a2 T λ − a2 Wt ).
Then the process W S = (WtS )t∈[0,T ] with
WtS
= Wt −
Zt
ηsS ds
0
is a d-dimensional standard Brownian motion with respect to G.
Proof. We use Lévy’s theorem characterizing Brownian motions (compare Theorem 39 in Protter
[1990]) in order to prove the theorem, i.e. we show:
(i) W S is a continuous G-martingale;
(ii) the quadratic variation is given by [W·S , W·S ]t = tId .
93
5 Continuous-Time Consumption-Investment Problems
(i) W S is a martingale w.r.t. the enlarged background filtration:
We consider Gs -measurable random variables of the form Zs = 1A∩C , where A is an Fs -measurable
set and C is measurable w.r.t. σ(ITS ), i.e. C = [ITS ∈ B] for a some set B ∈ B(Rd ). For s ≤ t we
show:
E[Zs (WtSk − WsSk )] = 0,
k = 1, . . . , d.
(5.4)
Then by using a monotone class argument we conclude that (5.4) holds for all bounded and Gs measurable random variables (i.e. W S is a martingale w.r.t. G).
In order to show (5.4) we use the following properties of the insider information ITS :
(1) We determine the conditional distribution of ITS given Ft . Therefore, we look at the conditional characteristic function of ITS . Let p ∈ Rd . Then
h ⊤ S
i
h ⊤
i
φI S (p) := E eip IT | Ft =E eip (a(λT +WT )+(1−a)ε) | Ft
T
h ⊤
i
=E eip (aλT +a(WT −Wt )+aWt +(1−a)ε) | Ft .
The increment WT − Wt is independent of Ft . Moreover, W, ε are independent and λ, Wt
are measurable w.r.t. Ft . Hence
h ⊤ S
i
h ⊤
i h ⊤
i
⊤
E eip IT | Ft =eip (aλT +aWt ) E eip a(WT −Wt ) | ✚
F
✚t E eip (1−a)ε | ✚
F
✚t
⊤
=eip
(aλT +aWt )− 21 p⊤ Kt p
,
d
i.e. ITS |Ft = N (aλT + aWt , Kt ).
(2) We consider the conditional density of ITS given Ft . Let x ∈ Rd . Then the conditional
density can be written as the following product
⊤ −1
1
1
e− 2 (x−(aλT +aWt )) Kt (x−(aλT +aWt ))
p
(2π)
det(Kt )
⊤ −1
1
1
e− 2 (x−aλ0 T ) A (x−aλ0 T )
=
d p
(2π) 2 det(A)
|
{z
}
d
2
(5.5)
=:fI S (x)
T
×
s
det(A) − 1 (x−(aλT +aWt ))⊤ K −1 (x−(aλT +aWt ))+ 1 (x−aλ0 T )⊤ A−1 (x−aλ0 T )
2
t
e 2
,
det(Kt )
|
{z
}
=:M x (t,Wt )
where A := a2 T 2 Σ0 + K0 . Here fI S denotes the density of ITS .
T
(3) Now we show that M x = (M x (t, Wt ))t∈[0,T ] is an F-martingale. Firstly, we apply Itô’s
formula (compare e.g. Itô’s Formula in Protter [1990]) in order to verify that M x is an
F-local martingale. Thus, we determine the relevant partial derivatives:
d
1
1
∂ x
M (t, Wt ) = − M x (t, Wt )
det(Kt )
∂t
2
det(Kt ) dt
1
d
− M x (t, Wt )(x − (aλT + aWt ))⊤ Kt−1 (x − (aλT + aWt ))
2
dt
1
= a2 M x (t, Wt )tr(Kt−1 )
2
1
− a2 M x (t, Wt )(x − (aλT + aWt ))⊤ Kt−1 Kt−1 (x − (aλT + aWt )).
2
94
5.2 Enlargement and Filtering
For the last equality we use Lemma A.4.1 for the derivative of the inverse of a matrix∂
M x (t, Wt ) the derivative of M x (t, Wt ) w.r.t. the k-th
valued function. We denote by ∂w
k
component of the Brownian motion W . Then
∂
x
M (t, Wt )
=a(x − (aλT + aWt ))⊤ Kt−1 M x (t, Wt )
∂wk
k=1,...,d
and the second derivative
∂2
x
M (t, Wt )
∂wk ∂wj
j,k=1,...,d
= −a2 Kt−1 M x (t, Wt ) + a2 Kt−1 (x − (aλT + aWt ))(x − (aλT + aWt ))⊤ Kt−1 M x (t, Wt ).
Note that due to the independence of Wtk and Wtj for k , j the quadratic variation of W is
given by [W·j , W·k ]t = δjk t (δjk = 1 if j = k and δjk = 0 otherwise). Finally, applying Itô’s
formula shows that M x is a local martingale
d
dM x (t, Wt ) =
X ∂
∂ x
M (t, Wt )dt +
M x (t, Wt )dWtk
∂t
∂wk
k=1
d
∂2
1 X
M x (t, Wt )d[W·j , W·k ]t
+
2
∂wk ∂wj
k,j=1
∂
∂ x
x
M (t, Wt )
dWt
= M (t, Wt )dt +
∂t
∂wk
k=1,...,d
!
∂2
1
x
M (t, Wt )
dt
+ tr
2
∂wk ∂wj
j,k=1,...,d
=M x (t, Wt )(Kt−1 (ax − (a2 λT + a2 Wt )))⊤ dWt ,
where we use
Since
d
RT P
0 k=1
a2 M x (t, Wt )(x − (aλT + aWt ))⊤ Kt−1 Kt−1 (x − (aλT + aWt ))
=tr a2 M x (t, Wt )(x − (aλT + aWt ))⊤ Kt−1 Kt−1 (x − (aλT + aWt ))
=a2 M x (t, Wt )tr Kt−1 (x − (aλT + aWt ))(x − (aλT + aWt ))⊤ Kt−1 .
2
E[ Kt−1 (ax − (a2 λT + a2 Wt )) k M x (t, Wt ) ]dt < ∞ for k = 1, . . . , d (Lemma
A.1.1) the local martingale M x is even a true martingale. Here (x)k denotes the k-th entry
of the d-dimensional vector x.
(4) Applying the product rule to M x (t, Wt )Wtk , k = 1, . . . , d, we obtain
d(M x (t, Wt )Wtk ) =Wtk dM x (t, Wt ) + M x (t, Wt )dWtk + d[M x (·, W· ), W·k ]t
=M x (t, Wt ) Kt−1 (ax − (a2 λT + a2 Wt )) k dt
+ Wtk M x (t, Wt )(Kt−1 (ax − (a2 λT + a2 Wt )))⊤ dWt + M x (t, Wt )dWtk .
(5) Let ek denote the k-th d-dimensional unit vector. As in (3) it follows that the F-local
martingale
Zt
0
⊤ k
−1
2
2
M x (u, Wu ) e⊤
dWu
k + Wu Ku (ax − (a λT + a Wu ))
95
5 Continuous-Time Consumption-Investment Problems
is even a true martingale, i.e. for all A ∈ Fs it holds:
E[1A
Zt
s
⊤ k
−1
2
2
M x (u, Wu ) e⊤
dWu ] = 0.
k + Wu Ku (ax − (a λT + a Wu ))
Let now Zs = 1A∩C with A ∈ Fs , C ∈ σ(ITS ), i.e. C = [ITS ∈ B] for some B ∈ B(Rd ). Then for
s ≤ t we obtain by the definition of W S
E[Zs (WtSk − WsSk )]
=E[1A 1C (Wtk − Wsk −
Zt
ηuSk du)]
s
=E[E[1A 1C (Wtk − Wsk −
Zt
s
ηuSk du) | Ft ]]
=E[1A (Wtk − Wsk )E[1C | Ft ]] − E[1A 1C
Zt
ηuSk du].
s
Note that E[1C | Ft ] = P(C | Ft ) = P(ITS ∈ B | Ft ). By Fubini’s theorem and using properties
(2)-(5) we get
E[Zs (WtSk − WsSk )]
Zt
Z
x
k
k
E[1A (Wt − Ws ) M (t, Wt )fI S (x)dx] − E[1A 1C ηuSk du]
(2)
=
T
s
B
(3)
(Wtk M x (t, Wt ) − Wsk M x (s, Ws ))fI S (x)dx] − E[1A 1C
Zt
ηuSk du]
E[1A (Wtk M x (t, Wt ) − Wsk M x (s, Ws ))]fI S (x)dx − E[1A 1C
Zt
ηuSk du]
=
E[1A
=
Z
Z
T
B
T
B
Z
(4)(5)
=
E[1A
=
E[1A
Ku−1 (ax − (a2 λT
s
B
(2)
Zt
Zt
s
s
s
+ a Wu )) k M x (u, Wu )du]fI S (x)dx − E[1A 1C
2
T
E[1B (ITS ) Ku−1 (aITS − (a2 λT + a2 Wu )) | Fu ]du] − E[1A 1C
k
Zt
Zt
ηuSk du]
s
ηuSk du]
s
Since A ∈ Fs ⊆ Fu for s ≤ u ≤ t assertion (5.4) follows by applying again Fubini’s theorem
E[Zs (WtSk − WsSk )]
=E[
=
E[1A 1C
E[E[1A 1C
s
Zt
s
96
Zt
Ku−1 (aITS
Zt
− (a λT + a Wu )) | Fu ]du] − E[1A 1C ηuSk du]
2
2
k
s
Zt
−1
S
2
2
Ku (aIT − (a λT + a Wu )) | Fu ]]du − E[1A 1C ηuSk du] = 0.
k
s
(5.6)
5.2 Enlargement and Filtering
We define the sets Hs,t and Ms as follows:
n
o
Hs,t := H R-valued, bounded and Gs − measurable : E[H(WtSk − WsSk )] = 0, k = 1, . . . , d ;
Ms :={1A∩C : A ∈ Fs and C ∈ σ(ITS )}.
Then Hs,t is a monotone vector space and Ms is a multiplicative class with σ(Ms ) = Gs (see
Lemma A.1.3). Due to (5.6) it holds that Ms ⊆ Hs,t . Hence, by Theorem 8 (monotone class
theorem) in Protter [1990] it follows that Hs,t contains all bounded and Gs -measurable random
variables (i.e. E[Zs (WtSk − WsSk )] = 0 for all bounded and Gs -measurable random variables Zs ).
Consequently, W Sk , k = 1, . . . , d, is a martingale w.r.t. G.
(ii) The quadratic variation of W S is given by [W·S , W·S ]t = tId :
The quadratic variation is a path property. Since Wtj , Wtk are independent for k , j and
is of finite variation on [0, T ], we have [W·Sk , W·Sj ]t = [W·k , W·j ]t = δjk t.
Rt
0
ηuSk du
All in all, it follows by Lévy’s characterization of a Brownian motion that W S is a G-Brownian
motion.
(II) Brownian Information
Let the investor have Brownian information, i.e. IT = ITW . For a known drift λ the σ-fields
generated by ITS = aYT + (1 − a)ε and ITW = aWT + (1 − a)ε are identical. Hence,
Gt := Ft ∨ σ(ITW ) = Ft ∨ σ(ITS )
(λ is measurable with respect to F0 ). Consequently, we obtain from Theorem 5.2.2:
Theorem 5.2.3. Let the G-adapted process η W = (ηtW )t∈[0,T ] be given by
ηtW := Kt−1 (aITW − a2 Wt ).
Then the process W W = (WtW )t∈[0,T ] with
WtW
= Wt −
Zt
ηsW ds
0
is a d-dimensional standard Brownian motion with respect to G.
Remark 5.2.4. Note that
ηtW = Kt−1 (aITW − a2 Wt ) = Kt−1 (aITS − a2 T λ − a2 Wt ) = ηtS ,
i.e. WtW = WtS .
(III) Drift Information
For drift information, i.e. IT = ITλ , the enlargement step is not needed. The exact drift information
ITλ = λ is already included in the background filtration F. Moreover, since the noise ε is independent
of the Brownian motion and independent of the drift, ITλ does not provide any (useful) additional
information. In this case the F-Brownian motion W is still a Brownian motion with respect to G
with
Gt := Ft ∨ σ(ITλ ).
97
5 Continuous-Time Consumption-Investment Problems
Theorem 5.2.5. The F-Brownian motion W is a d-dimensional standard Brownian motion with
respect to the enlarged background filtration G.
Proof. Using again Lévy’s characterization of a Brownian motion the assertion follows immediately. Let Zs = 1A∩C , where A is an Fs -measurable set and C is measurable w.r.t. σ(ITλ ), i.e.
Zs is Gs -measurable. Then for s ≤ t it holds
E[Zs (Wt − Ws )] = E[1A E[1C | Ft ] E[Wt − Ws | Fs ]] = 0.
|
{z
}
=0
For the first equality we use that E[1C | Ft ] is F0 -measurable. This can easily be seen by the
conditional distribution of ITλ given Ft which is independent of t
⊤ λ
IT
E[eip
⊤
| Ft ] = eip
aλ
⊤
E[eip
(1−a)ε
⊤
|✚
F
✚t ] = eip
aλ− 12 p⊤ (1−a)2 Σε p
for p ∈ Rd . Trivially, the quadratic variation is given by [W· , W· ]t = tId . By the same arguments
as in Theorem 5.2.2 it follows that W is a G-Brownian motion.
In case of drift information we can omit the enlargement step. Hence, we only have to estimate
the unknown drift on the basis of the observations and the drift information (see Section 5.2.2).
Insider Market
By replacing W in the observation process (5.1) by its G-semimartingale decomposition as given
in Theorem 5.2.2, Theorem 5.2.3 and Theorem 5.2.5 we get immediately:
Theorem 5.2.6. The observation process Y admits with respect to the enlarged filtration G the
following semimartingale decomposition
dYt = (λ + ηtIT )dt + dWtIT ,
where ηtIT and WtIT are given in Table 5.1.
Type of
insider
information
Insider
drift
ηtIT
Brownian
motion
WtIT
IT = ITS
Kt−1 (aITS − a2 T λ − a2 Wt )
Rt
Wt − Ks−1 (aITS − a2 T λ − a2 Ws )ds
ITW
Kt−1 (aITW
IT = ITλ
0
IT =
− a2 W
t)
0
Rt
Wt − Ks−1 (aITW − a2 Ws )ds
0
Wt
Table 5.1: Information drift and G-Brownian motion
If we replace λt + Wt in the stock price process by the G-semimartingale decomposition of Y (see
Theorem 5.2.6), then we get the dynamics of the stock price process S in the insider market
(i.e. with information filtration G), namely
(5.7)
dSt = diag(St1 , . . . , Std )σ (λ + ηtIT )dt + dWtIT .
98
5.2 Enlargement and Filtering
This equation represents the semimartingale decomposition of the stock price process with respect
to the insider filtration G. We call
λ + ηtIT
the insider drift. Analogously, we rewrite the wealth SDE so that
dXtπ
=
(Xtπ αt⊤ (σ(λ + ηtIT ) − r1d ) + rXtπ − ct )dt + Xtπ αt⊤ σdWtIT .
(5.8)
Remark 5.2.7. For the three types of insider information there always exist a conditional density
of IT given Ft which can be written as a product fIT (x)M x (t, Wt ) (compare (5.5)). Moreover,
the information drift ηtIT k , k = 1, . . . , d, is given by the finite variation part of the semimartingale
(Wtk M x (t, Wt ))t∈[0,T ] at x = IT .
So far, we included the insider information into the stock price process and rewrote the wealth
SDE. The stochastic integral occurring in (5.2) is in (5.8) now well-defined as a G-Brownian
integral: We integrate a G-adapted stochastic process with respect to a G-Brownian motion.
5.2.2 Filtering
Additionally to the current wealth the investor knows the value iT of the insider information
IT . Hence, (Xtπ , IT ) describes the state of the investor. However, the insider drift λ + η IT =
(λ + ηtIT )t∈[0,T ] and W IT are not adapted with respect to the information filtration FIT Y . The
insider drift is thus not observable to the insider and has to be estimated. We follow here the
approach of Danilova et al. [2010], i.e. we consider λ + η IT as the unknown drift which has to be
estimated on the basis of the observations and the insider information. By saying "estimated on
the basis of the information" we simply mean that the estimator is measurable with respect to
the σ-field generated by the information. The conditional expectation
E[λ + ηtIT | FtIT Y ]
is the best estimator of λ + ηtIT based on the information of the insider. "Best" in the sense that
the mean square error is minimized (see e.g. Liptser & Shiryaev [2001b] Chapter 12). Determining
this estimator represents a filtering problem. In the theory of stochastic filtering λ + ηtIT is
often called the signal and (Yt , IT ) the observation. Loosely speaking, we want to "filter" the
signal out of the observation and away from the "noise".
The following theorem is a crucial result in the filtering theory:
Theorem 5.2.8. The process W IT Y = (WtIT Y )t∈[0,T ] defined by
WtIT Y
:= Yt −
Zt
0
E[λ + ηsIT | FsIT Y ]ds
is a d-dimensional (standard) Brownian motion with respect to the insider information filtration
FIT Y .
Proof. We use Lévy’s characterization of a Brownian motion in order to prove this theorem, i.e.
we show that W IT Y is a continuous FIT Y -martingale and the quadratic variation is given by
99
5 Continuous-Time Consumption-Investment Problems
[W·IT Y , W·IT Y ]t = tId . Due to its definition W IT Y is adapted w.r.t FIT Y . Let s ≤ t. By the
definition of W IT Y and using FsIT Y ⊆ Gs it holds
h
i
E WtIT Y − WsIT Y | FsIT Y
 t

Z
| Gs ] | FsIT Y ] + E  λ + ηuIT − E[λ + ηuIT | FuIT Y ]du | FsIT Y  .
}
= E[E[WtIT − WsIT
|
{z
s
=0
By Jensen’s inequality, the tower property of the conditional expectation and the normal distribution of λ + ηtS (see (A.2)) it can easily be seen that the quantity
ZT
0
E[|λ + ηtSk − E[λ + ηtSk | FtIT Y ]|]dt, k = 1, . . . , d,
is finite. Thus, we are able to apply Fubini’s theorem for conditional expectations (see Lemma
A.3.2) to interchange the order of the integral and the conditional expectation
E
h
WtIT Y
− WsIT Y
|
FsIT Y
i
Zt h
i
= E λ + ηuIT Y − E[λ + ηuIT Y | FuIT Y ] | FsIT Y du
s
=
Zt
s
E[λ + ηuIT Y | FsIT Y ] − E[E[λ + ηuIT Y | FuIT Y ] | FsIT Y ] du = 0,
|
{z
}
I Y
=E[λ+ηuT
i.e. W IT Y is a martingale w.r.t. FIT Y . Since the process
Rt
0
finite variation on [0, T ] we have
I Y
|Fs T
]
λ + ηsIT Y − E[λ + ηsIT Y | FsIT Y ]ds is of
[W·IT Y , W·IT Y ]t = [W·IT , W·IT ]t = tId .
Using Lévy’s characterization completes the proof.
(I) Stock Price Information
Let IT = ITS . Applying Itô’s formula to the insider drift gives us the following dynamics of the
signal
d(λ + ηtS ) =d Kt−1 (aITS − a2 T λ − a2 Wt )
=a2 Kt−1 ηtS dt − a2 Kt−1 dWt
= − a2 Kt−1 dWtS .
(5.9)
For the last equality we use Theorem 5.2.2. The 2d-dimensional observation consists of the
additional information ITS ∈ Rd and the observation Yt ∈ Rd . Using the G-semimartingale decomposition of Y we end up with following filtering problem
100
5.2 Enlargement and Filtering
Signal
= −a2 Kt−1 dWtS ,
d(λ + ηtS )
λ + η0S = λ + K0−1 (aITS − a2 T λ)
Observation
dYt
dITS
=
=
(λ + ηtS )dt + dWtS
0.
We want to apply classical results from the theory of stochastic filtering. In order that the given
filtering problem fits into the framework of Theorem 12.7 in Liptser & Shiryaev [2001b], we
slightly modify the second component of the observation (insider information)
Itc = ITS + cW̄tS
(I0c = ITS ),
where W̄ S = (W̄tS )t∈[0,T ] is a d-dimensional G-Brownian motion independent of λ, ITS and W S ,
and c > 0 is a constant. Using matrix notation the modified filtering problem is of the following
form:
Signal
d(λ + ηtS ) =
dYt
dItc
−a2 Kt−1
dWtS
,
0d×d
dW̄tS
λ + η0S = (1 − a)2 Σε K0−1 λ + aK0−1 ITS
Observation
Id
Id
S
=
(λ + ηt )dt +
0d×d
0d×d
0d×d
cId
dWtS
,
dW̄tS
Y0
I0c
0d×1
.
=
ITS
We denote the filtration generated by the observation by
FtI
c
Y
:= σ(Iuc , Yu : 0 ≤ u ≤ t).
Theorem 5.2.9. For t ∈ [0, T ] the estimator
E[λ + ηtS | FtI
c
Y
]
and the conditional covariance matrix
γt := E[(λ + ηtS − E[λ + ηtS | FtI
c
c
Y
])(λ + ηtS − E[λ + ηtS | FtI
c
Y
])⊤ | FtI
c
Y
]
are unique continuous FtI Y -measurable solutions to
c
c
d E[λ + ηtS | FtI Y ] =(−a2 Kt−1 + γt )dYt + (a2 Kt−1 − γt )E[λ + ηtS | FtI Y ]dt;
(5.10)
γ̇t =a2 γt Kt−1 + a2 Kt−1 (γt )⊤ − γt γt⊤
with initial conditions
E[λ + η0S | F0I
c
Y
] =(1 − a)2 Σε (K0 + a2 T 2 Σ0 )−1 λ0 + a(T Σ0 + Id )(K0 + a2 T 2 Σ0 )−1 ITS ;
γ0 =(1 − a)4 Σε (K0 + a2 T 2 Σ0 )−1 Σ0 K0−1 Σε .
101
5 Continuous-Time Consumption-Investment Problems
Remark 5.2.10. γt is a d × d-matrix and its derivative γ̇t is defined componentwise. Since γ0
is positive definite it follows from Theorem 12.7 in Liptser & Shiryaev [2001b] that γt is also
positive definite.
Proof. We apply Theorem 12.7 in Liptser & Shiryaev [2001b]. By Lemma A.1.2 the conditions
c
(1) − (10) of Theorem 12.7 are satisfied. It follows that E[λ + ηt | FtI Y ] and γt are unique conc
tinuous FtI Y -measurable solutions of the system of equations given in (5.10). For the initial
conditions we consider
φS := (λ − λ0 ) − aT Σ0 (K0 + a2 T 2 Σ0 )−1 (ITS − aλ0 T ).
The random variables φS and ITS are independent (see Lemma 4.3.14). We rewrite λ + η0S in
terms of ITS and φS , i.e.
λ + η0S =(1 − a)2 Σε K0−1 λ + aK0−1 ITS
=(1 − a)2 Σε K0−1 φS + (1 − a)2 Σε (K0 + a2 T 2 Σ0 )−1 λ0 + a(T Σ0 + Id )(K0 + a2 T 2 Σ0 )−1 ITS .
Then for p ∈ Rd we obtain by the independence of ITS and φS
⊤
E[eip
(λ+η0S )
| F0I
c
Y
]
= exp(ip⊤ ((1 − a)2 Σε (K0 + a2 T 2 Σ0 )−1 λ0 + a(T Σ0 + Id )(K0 + a2 T 2 Σ0 )−1 ITS ))
× E[exp(ip⊤ (1 − a)2 Σε K0−1 φS )]
⊤
=eip
S
((1−a)2 Σε (K0 +a2 T 2 Σ0 )−1 λ0 +a(T Σ0 +Id )(K0 +a2 T 2 Σ0 )−1 IT
)
1 ⊤ (1−a)4 Σ (K +a2 T 2 Σ )−1 Σ K −1 Σ p
ε
ε
0
0
0 0
× e− 2 p
d
since φS = N 0, K0 (K0 + a2 T 2 Σ0 )−1 Σ0 .
Lemma 5.2.11. The estimator of the modified filtering problem coincides with the estimator of
the original filtering problem, i.e.
E[λ + ηtS | FtI
c
Y
] = E[λ + ηtS | FtIT Y ]
for all t ∈ [0, T ].
Remark 5.2.12. The conditional covariance matrix of the modified filtering problem coincides
with the conditional covariance matrix of the original filtering problem. Moreover, since γt is
deterministic its trace corresponds to the mean square error.
S
S
Proof. We consider the filtration FIT Y W̄ = (FtIT Y W̄ )t∈[0,T ] with
FtIT Y W̄
S
:= σ(ITS , Yu , W̄uS : 0 ≤ u ≤ t).
Obviously, it holds that (see definition of Itc with I0c = ITS )
FtI
c
Y
S
= FtIT Y W̄ ,
t ∈ [0, T ].
Since ITS , Y and λ + η S are independent of W̄ S we have
E[λ + ηtS | FtI
c
Y
S
] = E[λ + ηtS | FtIT Y W̄ ] = E[λ + ηtS | FtIT Y ].
102
5.2 Enlargement and Filtering
Theorem 5.2.13. Let the symmetric matrix ΣS
a be given by
2 −1
4
2 2
−1
ΣS
Σ0 K0−1 Σε
a := −a K0 + (1 − a) Σε (K0 + a T Σ0 )
and let (Id + ΣS
a t) be non-singular for all t ∈ [0, T ]. Then it holds:
a) The estimator E[λ + ηtS | FtIT Y ] has the following explicit form
−1
E[λ + ηtS | FtIT Y ] = (Id + ΣS
(E[λ + η0S | F0IT Y ] + ΣS
a t)
a Yt ),
where
E[λ + η0S | F0IT Y ] = (1 − a)2 Σε (K0 + a2 T 2 Σ0 )−1 λ0 + a(T Σ0 + Id )(K0 + a2 T 2 Σ0 )−1 ITS .
The conditional covariance matrix γt is given by
−1 S
γt = (Id + ΣS
Σa + a2 Kt−1 .
a t)
b) The dynamics of the estimator E[λ + ηtS | FtIT Y ] are given by
−1 S
d E[λ + ηtS | FtIT Y ] = (Id + ΣS
Σa dWtSY .
a t)
Remark 5.2.14. The optimal estimator is linear, i.e. the latter depends linearly on the observation (ITS , Yt ). For Kalman-Bucy filtering problems it is well-known that the best linear estimator
is also the optimal filter (see for instance Øksendal [2010]).
Proof.
a) Let
−1 S
γt := (Id + ΣS
Σa + a2 Kt−1
a t)
(5.11)
and note that γt is symmetric. Then
−1 S
−1 S
γ̇t = − (Id + ΣS
Σa (Id + ΣS
Σa + a4 Kt−1 Kt−1
a t)
a t)
=a2 γt Kt−1 + a2 Kt−1 γt − γt γt ,
i.e. γt solves the (matrix-valued) ordinary differential equation (ODE) given in Theorem
5.2.9. The process Y is an FIT Y -semimartingale (see Theorem 5.2.8). By applying Itô’s
−1 (E[λ + η S | F IT Y ] + ΣS Y ) we get
formula to the semimartingale f (t, Yt ) := (Id + ΣS
a t)
a t
0
0
−1 S
−1
df (t, Yt ) = − (Id + ΣS
Σa (Id + ΣS
(E[λ + η0S | F0IT Y ] + ΣS
a Yt )dt
a t)
a t)
−1 S
+ (Id + ΣS
Σa dYt
a t)
−1 S
−1 S
= − (Id + ΣS
Σa f (t, Yt )dt + (Id + ΣS
Σa dYt
a t)
a t)
=(−a2 Kt−1 + γt )dYt + (a2 Kt−1 − γt )f (t, Yt )dt.
(5.12)
Since the filter equation has a unique continuous FtIT Y -measurable solution (compare Theorem 5.2.9) it follows that
E[λ + ηtS | FtIT Y ] = f (t, Yt )
103
5 Continuous-Time Consumption-Investment Problems
b) Using the FIT Y -semimartingale decomposition of Y (see Theorem 5.2.8) the assertion follows
by (5.12).
Remark 5.2.15. Whereas we consider λ + ηtS as the unknown drift and determine its estimator
E[λ + ηtS | FtIT Y ], Hansen [2013] considers the (initial) unknown parameter λ as the drift which
has to be estimated on the basis of the insider information and the observations. Hence, the
filtering problem of Hansen [2013] consists of determining E[λ | FtIT Y ] and has the following form
Signal
dλ
=
0
Observation
dYt
=
(1 − a)2 Kt−1 Σε λdt + aKt−1 ITS dt − a2 Kt−1 Yt dt + dWtS
dITS
=
0,
where the finite variation part of the G-semimartingale Y is decomposed into a "λ-part", an "ITS part" and a "Y -part". The connection between the two different filtering problems is obvious
E[λ + ηtS | FtIT Y ] = (1 − a)2 Kt−1 Σε E[λ | FtIT Y ] + Kt−1 (aITS − a2 Yt ).
(II) Brownian Information
Now we consider an investor with Brownian information, i.e. IT = ITW . Applying Itô’s formula
to the insider drift (with Brownian information) gives us the following dynamics of the signal
d(λ + ηtW ) = d Kt−1 (aITW − a2 Wt ) = −a2 Kt−1 dWtW .
Since we follow here the same approach as with stock price information we just summarize the
results. In order to formulate the modified filtering problem we first define the modified insider
information
Itc = ITW + cW̄tW
(I0c = ITW ),
where c > 0 is a constant and W̄ W = (W̄tW )t∈[0,T ] is a d-dimensional G-Brownian motion independent of λ, IT and W W . Hence, the modified filtering problem is given by:
Signal
d(λ + ηtW )
dYt
dItc
=
−a2 Kt−1
dWtW
,
0d×d
dW̄tW
Observation
Id
Id
W
=
(λ + ηt )dt +
0d×d
0d×d
λ + η0W = λ + aK0−1 ITW
0d×d
cId
dWtW
dW̄tW
,
Y0
I0c
0d×1
=
.
ITW
We again obtain by Theorem 12.7 in Liptser & Shiryaev [2001b] the following result (for the
initial conditions compare Lemma 4.3.16).
104
5.2 Enlargement and Filtering
Theorem 5.2.16. For t ∈ [0, T ] the estimator E[λ + ηtW | FtIT Y ] and the conditional covariance
matrix γt := E[(λ + ηtW − E[λ + ηtW | FtIT Y ])(λ + ηtW − E[λ + ηtW | FtIT Y ])⊤ | FtIT Y ] are unique
continuous FtIT Y -measurable solutions to
d E[λ + ηtW | FtIT Y ] =(−a2 Kt−1 + γt )dYt + (a2 Kt−1 − γt )E[λ + ηtW | FtIT Y ]dt;
γ̇t =a2 γt Kt−1 + a2 Kt−1 (γt )⊤ − γt γt⊤
with initial conditions
E[λ + η0W | F0IT Y ] =λ0 + aK0−1 ITW ;
γ0 =Σ0 .
The solutions to the SDE and the ODE can be determined explicitly:
Theorem 5.2.17. Let the symmetric matrix ΣW
a be given by
2 −1
ΣW
a := −a K0 + Σ0
and let (Id + ΣW
a t) be non-singular for all t ∈ [0, T ]. Then it holds:
a) The estimator E[λ + ηtW | FtIT Y ] has the following explicit form
−1
(E[λ + η0W | F0IT Y ] + ΣW
E[λ + ηtW | FtIT Y ] = (Id + ΣW
a Yt ),
a t)
where
E[λ + η0W | F0IT Y ] = λ0 + aK0−1 ITW .
The conditional covariance matrix γt is given by
−1 W
γt = (Id + ΣW
Σa + a2 Kt−1 .
a t)
b) The dynamics of the estimator E[λ + ηtW | FtIT Y ] are given by
−1 W
Σa dWtW Y .
d E[λ + ηtW | FtIT Y ] = (Id + ΣW
a t)
(III) Drift Information
We consider an investor with drift information, i.e. IT = ITλ . As the approach is the same as
before we again only summarize the results. Let c > 0 be a constant and let W̄ λ = (W̄tλ )t∈[0,T ]
be a d-dimensional G-Brownian motion independent of λ, ITλ and W λ . With the modified insider
information
Itc = ITλ + cW̄tλ
(I0c = ITλ ),
we obtain the following filtering problem:
dλ
dYt
dItc
=
Signal
0d×1
Observation
Id
Id
=
λdt +
0d×d
0d×d
0d×d
cId
dWt
,
dW̄tλ
Y0
I0c
0d×1
=
.
ITλ
105
5 Continuous-Time Consumption-Investment Problems
Solving the filtering problem we obtain (for the initial conditions compare Lemma 4.3.18):
Theorem 5.2.18. For t ∈ [0, T ] the estimator E[λ | FtIT Y ] and the conditional covariance matrix γt := E[(λ − E[λ | FtIT Y ])(λ − E[λ | FtIT Y ])⊤ | FtIT Y ] are unique continuous FtIT Y -measurable
solutions to
d E[λ | FtIT Y ] =γt dYt − γt E[λ | FtIT Y ]dt;
γ̇t = − γt γt⊤
with initial conditions
E[λ | F0IT Y ] =(1 − a)2 Σε (a2 Σ0 + (1 − a)2 Σε )−1 λ0 + aΣ0 (a2 Σ0 + (1 − a)2 Σε )−1 ITλ ;
γ0 =(1 − a)2 Σε (a2 Σ0 + (1 − a)2 Σε )−1 Σ0 .
The explicit solution to the SDE and the ODE are given by:
Theorem 5.2.19. Let the symmetric matrix Σλ
a be given by
2
2
2
−1
Σλ
Σ0
a := (1 − a) Σε (a Σ0 + (1 − a) Σε )
and let (Id + Σλ
a t) be non-singular for all t ∈ [0, T ]. Then it holds:
a) The estimator E[λ | FtIT Y ] has the following explicit form
−1
E[λ | FtIT Y ] = (Id + Σλ
(E[λ | F0IT Y ] + Σλ
a Yt ),
a t)
where
E[λ | F0IT Y ] = (1 − a)2 Σε (a2 Σ0 + (1 − a)2 Σε )−1 λ0 + aΣ0 (a2 Σ0 + (1 − a)2 Σε )−1 ITλ .
The conditional covariance matrix γt is given by
−1 λ
γt = (Id + Σλ
Σa .
a t)
b) The dynamics of the estimator E[λ | FtIT Y ] are given by
−1 λ
d E[λ | FtIT Y ] = (Id + Σλ
Σa dWtλY .
a t)
5.2.3 Summary of the Enlargement and Filtering Results
In Theorem 5.2.22 we summarize the results obtained by the initial enlargement of the background
filtration and the stochastic filtering for all three types of insider information.
As in the discrete-time case let the following assumption always be satisfied:
Assumption 5.2.20. The matrix (Id + ΣIaT T ) is positive definite, where ΣIaT is given in Table
5.2.
Remark 5.2.21. Assumption 5.2.20 implies that (Id + ΣIaT t) is non-singular for all t ∈ [0, T ].
Moreover, we can equivalently formulate Assumption 5.2.20 as
1
dIkT > − ,
T
k∈{1,...,d}
min
where dIkT , k = 1, . . . , d, are the eigenvalues of the symmetric matrix ΣIaT .
106
5.2 Enlargement and Filtering
Type of
insider information
d × d-matrix
ΣIaT
IT = ITS
−1 Σ
−a2 K0−1 + (1 − a)4 Σε (a2 T 2 K0 + K0 Σ−1
ε
0 K0 )
IT = ITW
−a2 K0−1 + Σ0
(1 − a)2 Σε (a2 Σ0 + (1 − a)2 Σε )−1 Σ0
IT = ITλ
Table 5.2: "modified covariance matrix" ΣIaT (continuous-time)
Theorem 5.2.22. Let Assumption 5.2.20 be satisfied. The observation process Y admits the
following semimartingale decomposition with respect to FIT Y
dYt = (Id + ΣIaT t)−1 (CλIT λ0 + C IT IT + ΣIaT Yt )dt + dWtIT Y ,
where CλIT , C IT are given in Table 5.2.3 and ΣIaT is given in Table 5.2.
Type of
insider
information
d × d-matrix
C IT
d × d-matrix
CλIT
IT = ITS
a(Id + T Σ0 )(a2 T 2 Σ0 + K0 )−1
IT = ITW
aK0−1
(1 − a)2 Σε (a2 T 2 Σ0 + K0 )−1
IT =
ITλ
Id
aΣ0 (a2 Σ0 + (1 − a)2 Σε )−1
(1 − a)2 Σε (a2 Σ0 + (1 − a)2 Σε )−1
Table 5.3: Coefficients of λ0 and Yt in the insider drift
The first column of Table 5.4 summarizes our approach: It shows the semimartingale decompositions of the observation process Y with respect to the filtrations F, G and FIT Y (note that it is
always the same process (ω-wise)). We start with the semimartingale decomposition of Y with
respect to F. By the enlargement step we get its G-representation (compare Section 5.2.1) and
then by stochastic filtering we finally obtain its FIT Y -representation (compare Section 5.2.2).
Enlargement ⇒ Filtering
F
⇓
G
⇓
FIT Y
dYt = λdt + dWt
dYt = (ηtIT + λ)dt + dWtIT
dYt = E[λ + ηtIT | FtIT Y ]dt + dWtIT Y
Filtering ⇒ Enlargement
F
⇓
FY
⇓
FIT Y
dYt = λdt + dWt
dYt = E[λ | FtY ]dt + dWtY
dYt = (E[λ | FtY ] + η̃tIT )dt + dWtIT Y
Table 5.4: Semimartingale representation of Y (w.r.t. different filtrations)
Replacing in the stock prices the observation Y by its FIT Y -representation gives us the dynamics
107
5 Continuous-Time Consumption-Investment Problems
of the stock price processes in the market with information filtration FIT Y
dSt = diag(St1 , . . . , Std )σ λ̂(t, IT , Yt )dt + dWtIT Y ,
(5.13)
where
λ̂(t, IT , Yt ) := (Id + ΣIaT t)−1 (CλIT λ0 + C IT IT + ΣIaT Yt )
(5.14)
or equivalently
dλ̂(t, IT , Yt ) = (Id + ΣIaT t)−1 ΣIaT dWtIT Y ,
λ̂(0, IT , 0) = CλIT λ0 + C IT IT .
(5.15)
Here λ̂(t, IT , Yt ) is the estimator of the insider drift, i.e.
λ̂(t, IT , Yt ) = E[λ + ηtIT | FtIT Y ].
From (5.15) we see immediately that λ̂(t, IT , Yt ) is an FIT Y -martingale. Analogously, we can
rewrite the wealth SDE in terms of W IT Y
dXtπ = (Xtπ αt⊤ (σ λ̂(t, IT , Yt ) − r1d ) + rXtπ − ct )dt + Xtπ αt⊤ σdWtIT Y .
(5.16)
Eventually, all processes occurring in the wealth SDE (5.16) are adapted with respect to the information filtration FIT Y . Hence, we reduced (P c ) by including the partial and insider information
into the stock price process (see (5.13)) to a standard consumption-investment problem. The
"price" we pay for this reformulation is a stochastic time-dependent drift.
Remark 5.2.23. In our approach the enlargement step is followed by the filtering step (F ⇒
G ⇒ FIT Y ). However, the order of these two steps can also be interchanged as illustrated in the
second column of Table 5.4 (F ⇒ FY ⇒ FIT Y ). Hence, in the first step the partial information
is incorporated into the optimization problem by estimating the unknown drift λ only on the basis
of the observations Y. Thus, the filtering problem has the following form
Signal
dλ
dYt
=
0
Observation
= λdt + dWt .
By setting a = 0 in the filtering problems with insider information we obtain directly its solution
E[λ | FtY ]
γ̃t
=
(Id + Σ0 t)−1 (λ0 + Σ0 Yt );
:= E[(λ − E[λ | FtY ])(E[λ | FtY ])⊤ | FtY ] = (Id + Σ0 t)−1 Σ0 .
Furthermore, W Y = (WtY )t∈[0,T ] with
dWtY = dYt − E[λ | FtY ]dt
is a Brownian motion with respect to FY . Afterward, in the second step the insider information
is incorporated by the initial enlargement of FY by σ(IT ). Analogously to Section 5.2.1 it can be
shown that there exists an FIT Y -adapted stochastic process η̃ IT = (η̃tIT )t∈[0,T ] such that
dWtIT Y = dWtY − η̃tIT dt.
For instance, if IT = ITS , then the information drift η̃ IT is given by
η̃tIT = (Id + (T − t)γ̃t )(Kt + a2 (T − t)2 γ̃t )−1 (aITS − a2 (Yt + (T − t)E[λ | FtY ])).
Due to the uniqueness of the semimartingale decomposition of a continuous semimartingale we
finally end up with the same FIT Y -representation of Y
E[λ + ηtIT | FtIT Y ] = E[λ | FtY ] + η̃tIT .
108
5.3 Properties of the Information Market
5.3 Properties of the Information Market
We consider the market with information filtration FIT Y .
Definition 5.3.1. An admissible consumption-investment strategy π = (ct , αt )t∈[0,T ] is called an
arbitrage opportunity if the corresponding wealth process satisfies: X0π = 0,
P(XTπ ≥ 0)
and
P(XTπ > 0) > 0.
The financial market is said to be free of arbitrage if there does not exist an arbitrage opportunity.
Equivalent Martingale Measure and No Arbitrage
In this section about arbitrage and equivalent martingale measures we suppose following assumption to be satisfied.
Assumption 5.3.2. Let dIkT , 0 for k = 1, . . . , d, i.e. ΣIaT is non-singular.
Definition 5.3.3. A probability measure Q on (Ω, FTIT Y ) is called an equivalent martingale
measure if Q is equivalent to P (on FTIT Y ) and the discounted stock price processes
are local FIT Y -martingales under Q.
Stk
Bt , k = 1, . . . , d,
Our financial market with information filtration FIT Y is based on the Brownian motion W IT Y .
Since the volatility matrix σ is non-singular there exists at most one martingale measure Q on
(Ω, FtIT Y ) that is locally equivalent to P. If Q exists then it has the form
dQ = Lt , t ∈ [0, T ],
(5.17)
dP F IT Y
t
with density process
 t

Z
Lt = exp − (λ̂(s, IT , Ys ) − rσ −1 1d )dWsIT Y 
0

1
× exp −
2
Zt
0

(λ̂(s, IT , Ys ) − rσ −1 1d )⊤ (λ̂(s, IT , Ys ) − rσ −1 1d )ds .
By Itô’s formula L = (Lt )t∈[0,T ] satisfies the SDE
dLt = −Lt λ̂(t, IT , Yt ) − rσ −1 1d dWtIT Y ,
L0 = 1
(5.18)
(see Lemma A.1.6). L is obviously a non-negative FIT Y -local martingale and hence an FIT Y supermartingale. Applying Iô’s formula to the semimartingale
f (t, λ̂(t, IT , Yt ))
q
1
IT
−1
⊤
IT −1
−1
:= det(Id + Σa t) exp
(λ̂(0, IT , 0) − rσ 1d ) (Σa ) (λ̂(0, IT , 0) − rσ 1d )
2
1
−1
⊤
IT
IT −1
−1
× exp − (λ̂(t, IT , Yt )−rσ 1d ) (Id + Σa t)(Σa ) (λ̂(t, IT , Yt )−rσ 1d )
2
109
5 Continuous-Time Consumption-Investment Problems
shows that the solution to the SDE (5.18) is given by Lt = f (t, λ̂(t, IT , Yt )). Note that (ΣIaT )−1 is
well-defined due to Assumption 5.3.2. Using this explicit formula of Lt we show in Lemma A.1.7
that E[LT | F0IT Y ] = 1. Hence, the local martingale L is even a true FIT Y -martingale and Q is
indeed a probability measure. By means of Girsanov’s theorem the process W Q = (WtQ )t∈[0,T ]
with
dWtQ = dWtIT Y + (λ̂(t, IT , Yt ) − rσ −1 1d )dt,
W0Q = 0,
is an FIT Y -Brownian motion under the measure Q (see Lemma A.1.8).
Remark 5.3.4. If Assumption 5.2.20 is not satisfied, then λ̂(t, IT , Yt ) is in general not square
integrable on [0, T ]. This fact prevents us from doing a Girsanov measure change as in (5.17).
Theorem 5.3.5. The market with information filtration FIT Y is free of arbitrage.
Proof. According to the "First Fundamental Theorem of Asset Pricing" the market is free of
arbitrage if there exists a martingale measure equivalent to P. The proof is quite standard and
can in the case without consumption be found in Korn & Korn [2001]. Let us assume that there
exists an arbitrage opportunity, i.e. there exists an admissible strategy π = (ct , αt )t∈[0,T ] with
X0π = 0, XTπ ≥ 0 a.s. and P(XTπ > 0) > 0.
(5.19)
By Itô’s formula we get for the discounted wealth process (see Lemma A.1.9)
Xtπ
+
Bt
i.e. Mtπ :=
Xtπ
Bt
+
Rt
0
cs
Bs ds
Zt
0
cs
ds = x0 +
Bs
Zt
0
Xsπ ⊤
α σdWsQ ,
Bs s
is a local martingale under Q. Since π is an admissible strategy Mtπ is
non-negative. Hence, (Mtπ )t∈[0,T ] is an FIT Y -supermartingale under Q with
0 = M0π ≥ EQ [MTπ ].
Consequently Q(MTπ > 0) = 0 and thus P(MTπ > 0) = 0. But this is a contradiction to the arbitrage
assumption (5.19).
Information Equivalence and Martingale Representation
We consider the σ-fields FtIT W
IT Y
FtIT W
and FtIT Y , where
IT Y
:= σ(IT , WuIT Y : 0 ≤ u ≤ t) ∨ N .
The following Lemma states that the two σ-fields are the same. This indicates that the Brownian
motion W IT Y and the insider information IT contain the same information as the observation
consisting of IT and Y . Then W IT Y is also called innovation process.
Lemma 5.3.6. For all t ∈ [0, T ] it holds:
FtIT W
110
IT Y
= FtIT Y .
5.3 Properties of the Information Market
IT Y
Proof. From (5.15) it follows immediately that λ̂(t, IT , Yt ) is measurable with respect to FtIT W
IT Y
Rt
and thus Yt = λ̂(s, IT , Ys )ds + WtIT Y is also measurable with respect to FtIT W
. Hence,
0
FtIT Y ⊆ FtIT W
IT Y
. Since the reverse inclusion is obviously true, the two σ-fields are the same.
Remark 5.3.7. The information filtration is equal to the natural filtration of the FIT Y -Brownian
motion W IT Y initially enlarged by the insider information IT (and augmented by the P-null sets
I Y
I Y
of F). We denote the augmented Brownian filtration by FW T = (FtW T )t∈[0,T ] . Since W IT Y
I Y
is a Brownian motion with respect to FIT Y the σ-fields σ(IT ) and FtW T are independent. Due
to the right-continuity of the augmented Brownian filtration (see for instance Protter [1990]) it is
not difficult to show (using the independence) that FIT Y is also right-continuous (see Amendinger
[2000]).
For a more general setting a similar representation result as below is shown in Amendinger
[2000]. However, due to the information equivalence (Lemma 5.3.6) the proof simplifies here
considerably since we can use standard martingale representation results (for the augmented
Brownian filtration).
Theorem 5.3.8. Let M = (Mt )t∈[0,T ] be an R-valued right-continuous FIT Y -martingale with
sup E[|Mt |] < ∞.
t∈[0,T ]
Then there exists an FIT Y -progressively measurable process H = (Hs )s∈[0,T ] with
a.s. such that for t ∈ [0, T ]
Mt = M0 +
Z
0
t
RT
0
Hs⊤ Hs ds < ∞
Hs⊤ dWsIT Y ,
where M0 is an F0IT Y -measurable random variable.
Proof. We prove the Theorem in two steps:
(i) We first show the assertion for R-valued FIT Y -martingales satisfying sup E[Mt2 ] < ∞.
t∈[0,T ]
(ii) By localization the result obtained in (i) can then be extended to martingales satisfying the
weaker requirement sup E[|Mt |] < ∞.
t∈[0,T ]
We use the following abbreviations for L2 -spaces:
(
)
L2 (FT ) := X R-valued, FT -measurable random variable: E[X 2 ] < ∞
 T

)
Z
2,d
d
⊤
L ([0, T ], F) := X = (Xt )t∈[0,T ] R -valued, F-progressively measurable: E  Xs Xs ds < ∞ .
(
0
I Y
(i) By Lemma 5.3.6 the σ-field FTIT Y can be decomposed into the independent σ-fields FTW T
and σ(IT ). We follow the idea of Amendinger [2000] and define the vector space V
)
(
m
X
IT Y
k k
k
∞
W IT Y
k
∞
∞
X Y with X ∈ L (FT
), Y ∈ L (σ(IT )) .
V := Z ∈ L (FT ) : H =
k=1
111
5 Continuous-Time Consumption-Investment Problems
Here L∞ (FT ) is the space of essentially bounded FT -measurable random variables. According to Theorem 3.5.1 in Malliavin [1995] V is dense in L2 (FTIT Y ). Let Z ∈ L2 (FTIT Y ). Then
there exists a sequence (Z n )n∈N ∈ V with
m(n)
n
Z =
X
X (k,n) Y (k,n) ,
X (k,n) ∈ L∞ FTW
k=1
IT Y
(k,n)
,Y
∈ L∞ σ(IT )
I Y we are able to apply
such that E[(Z n − Z)2 ] −→ 0 as n → ∞. Since X (k,n) ∈ L∞ FTW T
standard martingale representation results, i.e. by Theorem 52 (the martingale represen(k,n)
tation) in Korn & Korn [2001], for instance, there exists a process ψ = (ψt
)t∈[0,T ] ∈
IT Y
2,d
W
such that
L
[0, T ], F
ZT
X (k,n) = E[X (k,n) ] +
(k,n) ⊤
) dWsIT Y
(ψs
(5.20)
0
I Y
(Note that Mt := E[X (k,n) | FtW T ] is a square integrable martingale. We thus get representation (5.20) by applying Theorem 52 on Mt ). Therefore
m(n)
n
Z =
X
E[X
(k,n)
]Y
(k,n)
+
k=1
ZT
(k,n) ⊤
(ψs
) dWsIT Y Y (k,n) .
0
By the independence of X (k,n) and σ(IT ) it holds
E[X (k,n) ]Y (k,n) = E[X (k,n) Y (k,n) | F0IT Y ].
Consequently, we obtain by Theorem 33 in Protter [1990]
T
n
n
Z = E[Z |
where ϕn
s :=
m(n)
P
(k,n)
Y (k,n) ψs
F0IT Y
]+
d Z
X
⊤
IT Y
,
(ϕn
s ) dWs
(5.21)
i=1 0
(Rd -valued). Since Z n converges in L2 to Z it is easy to see
k=1
by Jensen’s inequality that E[Z n | F0IT Y ] converges in L2 to E[Z | F0IT Y ] as n → ∞. Thus,
also the second sum in (5.21) converges in L2 . As L2 [0, T ], FIT Y is a Hilbert space (see
e.g. Pham [2009]) it follows by Itô’s isometry that
S :=
 T
Z

IT Y
φ⊤
s dWs
0


: φ = (φt )t∈[0,T ] ∈ L2,d ([0, T ], FIT Y )

has to be a closed subspace of L2 FTIT Y . Hence, there exists a process ϕ ∈ L2 [0, T ], FIT Y
such that
ZT
0
112
⊤
IT Y
(ϕn
s ) dWs
−→
ZT
0
ϕs dWsIT Y
5.4 Hamilton-Jacobi-Bellman Equation
in L2 as n → ∞. (5.21) converges therefore in L2 to
Z = E[Z
| F0IT Y
]+
ZT
ϕs dWsIT Y .
0
∈ L2,1 ([0, T ], FIT Y
For any martingale M
) the assertion follows by setting Z = MT and using
the martingale property of the Brownian integral for L2 -bounded integrands.
(ii) The result of part (i) can be extended to martingales which are not square integrable by
following the proof of Liptser & Shiryaev [2001a] Theorem 5.7. However, the standard martingale representation result (i.e. with augmented Brownian filtration) for square integrable
martingales used by Liptser & Shiryaev [2001a] has to be replaced by the representation
result of part (i). Note that Theorem 5.7 (in Liptser & Shiryaev [2001a]) is a representation
w.r.t. a one-dimensional Brownian motion. But without any difficulties the proof can be
extended to a representation w.r.t. a d-dimensional Brownian motion.
5.4 Hamilton-Jacobi-Bellman Equation
After the enlargement and filtering step in Section 5.2 the state of the investor can be described
by Z π = (Ztπ )t∈[0,T ] , where
 π  π ⊤

dXt
(Xt αt (σ λ̂(t, IT , Yt ) − r1d ) + rXtπ − ct )dt + Xtπ αt⊤ σdWtIT Y

0
dZtπ :=  dIT  = 
(5.22)
I
Y
T
dYt
λ̂(t, IT , Yt )dt + dWt
⊤
with initial state Z0π = x0 IT⊤ 0 . The estimator of the insider drift is here given by
λ̂(t, IT , Yt ) := (Id + ΣIaT t)−1 (CλIT λ0 + C IT IT + ΣIaT Yt ).
Thus, the state process Z π is a (1 + 2d)-dimensional process consisting of the wealth, the insider
information and the observation. Z π depends through the wealth process X π on the chosen and
admissible action π. Hence, we are able to control the state process. Moreover, Z π forms
a Markov process (see Remark 5.4.1) which contains all information which the investor needs
in order to make his investment and consumption decisions, i.e. according to (5.22) we reduced
the consumption-investment problem to a (standard) Markovian control problem with state
process (Xtπ , IT , Yt ) and with a maximal expected reward over [t, T ] for which holds
 T

Z
I
Y
sup Et(x,iT ,y)  Uc (cs )ds + Up (XTπ ) | Ft T  = V (t, x, iT , y)
π ∈At
t
with value function
V (t, x, iT , y) ≔ sup Et(x,iT
π ∈At

 T
Z
 Uc (cs )ds + Up (XTπ ) .
,y)
t
The notation Et(x,iT ,y) [·] is short for E[· | Xtπ = x, IT = iT , Yt = y]. Hence, V (t, x, iT , y) is the
maximal expected reward over the remaining time from t to T when (x, iT , y) is the state at time
t. Furthermore, the value V0∗ (x0 , iT ) of the consumption-investment problem (P c ) is given by
V0∗ (x0 , iT ) := V (0, x0 , iT , 0).
113
5 Continuous-Time Consumption-Investment Problems
There are several methods in order to solve such a (standard) consumption-investment problem,
e.g. the dual or martingale approach and the dynamic programming approach. For a brief
overview of the mentioned methods we refer to Korn & Korn [2001]. In Danilova et al. [2010] the
martingale method is used to solve a terminal wealth problem of an insider. However, in such a
Markovian setting it is convenient to use the dynamic programming approach (as also done in
Hansen [2013]).
Remark 5.4.1. The process Z π needs not to be a Markov process for any admissible strategy π.
However, Z π is a Markov process for Markovian strategies π and in settings as considered here the
expected utility cannot be improved by using non-Markovian strategies (see for instance Øksendal
[2010] Theorem 11.2.3). Hence, we only consider strategies π ∈ A0 which are Markovian.
The dynamic programming approach yields to a second-order and non-linear partial differential
equation (PDE), the known as Hamilton-Jacobi-Bellman (HJB) equation. For more details
on the dynamic programming approach and the deviation of the HJB equation we refer to Pham
[2009] (Chapter 3) and to Fleming & Soner [1993] (Chapter IV). Before we set up the HJB
equation corresponding to our optimization problem we introduce some abbreviations in order to
simplify the notations:

 ⊤
xu (σ λ̂(t, iT , y) − r1d ) + rx − c
 ∈ R1+2d ;
0
µ(t, x, iT , y, c, u) := 
λ̂(t, iT , y)
 ⊤ 
xu σ
σ(t, x, iT , y, c, u) :=  0  ∈ R(1+2d)×d .
Id
Then
dZtπ = µ(t, Xtπ , IT , Yt , ct , αt )dt + σ(t, Xtπ , IT , Yt , ct , αt )dWtIT Y
and the HJB equation with terminal condition v(T, x, iT , y) = Up (x) has the form
Uc (c) + vz (t, x, iT , y)⊤ µ(t, x, iT , y, c, u)
0 =vt (t, x, iT , y) + sup
c≥0,u∈Rd
1
⊤
,
+ tr vzz (t, x, iT , y)σ(t, x, iT , y, c, u)σ(t, x, iT , y, c, u)
2
where
vt (t, x, iT , y) :=
vz (t, x, iT , y) :=
vzz (t, x, iT , y) :=
vx (t, x, iT , y) :=
vxx (t, x, iT , y) :=
114
∂
v(t, x, iT , y);
∂t


vx (t, x, iT , y)
vi (t, x, iT , y) ;
T
vy (t, x, iT , y)

vxx (t, x, iT , y) vxiT (t, x, iT , y)⊤
vxi (t, x, iT , y) vi i (t, x, iT , y)
T
T T
vxy (t, x, iT , y)
viT y (t, x, iT , y)
∂
v(t, x, iT , y);
∂x
∂2
v(t, x, iT , y);
∂x∂x

vxy (t, x, iT , y)⊤
viT y (t, x, iT , y)⊤  ;
vyy (t, x, iT , y)
5.5 Solutions for Logarithmic and Power Utility Functions
vy (t, x, iT , y)
vxy (t, x, iT , y)
vyy (t, x, iT , y)

∂
∂y1 v(t, x, iT , y)


..
;
.

∂
v(t,
x,
i
,
y)
T
∂yd
 2

∂
∂x∂y1 v(t, x, iT , y)


..
;
:= 
.


2
∂
∂x∂yd v(t, x, iT , y)

2
∂2
v(t, x, iT , y) ∂y∂1 ∂y2 v(t, x, iT , y)
1
 ∂y∂1 ∂y
2
∂2

 ∂y2 ∂y1 v(t, x, iT , y) ∂y2 ∂y2 v(t, x, iT , y)
:= 
..

.

2
∂
v(t,
x, iT , y)
...
∂y ∂y1

:= 

d
...
...
..
.

∂2
∂y1 ∂yd v(t, x, iT , y)

∂2

∂y2 ∂yd v(t, x, iT , y)
∂2
∂yd ∂yd
.


v(t, x, iT , y)
Since the insider information does not change over time, the HJB equation of our consumptioninvestment problem simplifies to:
0 = vt (t, x, iT , y) +
sup
c≥0,u∈Rd
Uc (c) + vx (t, x, iT , y) xu⊤ (σ λ̂(t, iT , y) − r1d ) + rx − c
1
+ vy (t, x, iT , y)⊤ λ̂(t, iT , y) + vxx (t, x, iT , y)x2 u⊤ Σu + xu⊤ σvxy (t, x, iT , y)
2
!
1
+ tr (vyy (t, x, iT , y))
2
(5.23)
with terminal condition v(T, x, iT , y) = Up (x).
Note that the HJB equation contains first and second derivatives with respect to t, x and y, but
there are no derivatives with respect to iT .
Remark 5.4.2. If there exists a "smooth" solution to the HJB equation, then it follows by a
verification step that this smooth solution is the value function. However, the existence of a
smooth solution can in general not be ensured but one may hope for solutions in a "weaker" from so-called viscosity solutions. For the verification step and for viscosity solutions we refer to Pham
[2009] (Section 3.6 and Chapter 4).
5.5 Solutions for Logarithmic and Power Utility Functions
In this section we solve the consumption-investment problems for a continuous-time insider having
either logarithmic or power utility functions.
Recall that we denote by dIkT , k = 1, . . . , d, the eigenvalues of the symmetric matrix ΣIaT . Also
note that ΣIaT needs not to be non-singular such that there may exist eigenvalues with dIkT = 0
for some k ∈ {1, . . . , d}. We make use of the following abbreviation:
ΣIt T := (Id + ΣIaT t)−1 ΣIaT .
ΣIt T is symmetric due to the symmetry of ΣIaT .
115
5 Continuous-Time Consumption-Investment Problems
5.5.1 Logarithmic Utility Functions
We consider Uc (x) = Up (x) = log(x).
Theorem 5.5.1. Let Assumption 5.2.20 be satisfied. Furthermore, let without loss of generality
T
= . . . = dIdT = 0 for some j ∈ {0, . . . , d}. Then the function
dIj+1
v(t, x, iT , y) := (T − t + 1) log(x) + g(t, iT , y)
solves the HJB equation, where
j
1X
1
log
g(t, iT , y) := − (T − t + 1) log(T − t + 1) + j(T − t) +
2
2
k=1
1 + dIkT t
1 + dIkT T
!
1+
1 + dIkT T
dIkT
!
j
I
1 X dkT
1
+ (T − t)( (T − t) + 1) r +
2
2
1 + dI T t
k=1
k
!
⊤ −1
1
+ σ λ̂(t, iT , y) − r1d Σ
σ λ̂(t, iT , y) − r1d .
2
Proof. By solving the maximization problem in the HJB equation we get the candidate (optimal)
consumption strategy
C(t, x, iT , y) =
1
vx (t, x, iT , y)
(5.24)
and the candidate (optimal) investment strategy
Π(t, x, iT , y) = −Σ−1
vx (t, x, iT , y)(σ λ̂(t, iT , y) − r1d ) + σvxy (t, x, iT , y)
.
xvxx (t, x, iT , y)
(5.25)
Plugging in the candidate (optimal) strategy into the HJB equation gives the following second
order non-linear PDE
0 =vt (t, x, iT , y) − log(vx (t, x, iT , y)) + rxvx (t, x, iT , y) − 1
vx (t, x, iT , y) vx (t, x, iT , y)(σ λ̂(t, iT , y) − r1d )⊤
−
vxx (t, x, iT , y)
+ vxy (t, x, iT , y)⊤ σ Σ−1 (σ λ̂(t, iT , y) − r1d ) + vy (t, x, iT , y)⊤ λ̂(t, iT , y)
1
1
+
vx (t, x, iT , y)(σ λ̂(t, iT , y) − r1d )⊤ + vxy (t, x, iT , y)⊤ σ Σ−1
2 vxx (t, x, iT , y)
× vx (t, x, iT , y)(σ λ̂(t, iT , y) − r1d ) + σvxy (t, x, iT , y)
1
−
vx (t, x, iT , y)(σ λ̂(t, iT , y) − r1d )⊤ + vxy (t, x, iT , y)⊤ σ σ −1 vxy (t, x, iT , y)
vxx (t, x, iT , y)
1
(5.26)
+ tr (vyy (t, x, iT , y)) .
2
Motivated by the h-investor’s value function which can be separated into a sum, where the
first sum is independent of the observation and the insider information and the second one is
independent of the wealth, we are making the ansatz
v(t, x, iT , y) = f (t) log(x) + g(t, iT , y).
116
5.5 Solutions for Logarithmic and Power Utility Functions
f : [0, T ] → R and g : [0, T ] × Rd × Rd → R are functions which satisfy the terminal conditions
f (T ) = 1 and g(T, iT , y) = 0. The partial derivatives needed in the HJB equation are given by
vt (t, x, iT , y) = f ′ (t) log(x) + gt (t, iT , y)
f (t)
vx (t, x, iT , y) =
x
vy (t, x, iT , y) = gy (t, iT , y)
f (t)
vxx (t, x, iT , y) = − 2
x
vxy (t, x, iT , y) = 0d×1
vyy (t, x, iT , y) = gyy (t, iT , y).
Substituting the partial derivatives into (5.26) gives us
0
=
(f ′ (t) + 1) log(x) + rf (t) + gt (t, iT , y) − log(f (t)) − 1 + λ̂(t, iT , y)⊤ gy (t, iT , y)
1
1
+ tr (gyy (t, iT , y)) + f (t)(σ λ̂(t, iT , y) − r1d )⊤ Σ−1 (σ λ̂(t, iT , y) − r1d ).
(5.27)
2
2
Since g and f are independent of x and the PDE (5.27) holds for all (t, x, iT , y) ∈ [0, T ] × R × Rd ×
Rd we conclude f ′ (t) = −1. With the terminal condition f (T ) = 1 we then obtain
f (t) = T − t + 1.
We define
A(t, iT , y) :=r(T − t + 1) − log(T − t + 1) − 1
1
+ (T − t + 1)(σ λ̂(t, iT , y) − r1d )⊤ Σ−1 (σ λ̂(t, iT , y) − r1d ).
2
According to (5.27) it remains to solve the PDE
1
gt (t, iT , y) + λ̂(t, iT , y)⊤ gy (t, iT , y) + tr (gyy (t, x, iT , y)) = −A(t, iT , y)
2
(5.28)
with terminal condition g(T, iT , y) = 0. Motivated by the solutions (5.24) and (5.25) of the maximization problem within the HJB equation, we suppose
∗
αt∗ ≔Π(t, Xtπ , λ̂(t, IT , Yt )) = Σ−1 (σ λ̂(t, IT , Yt ) − r1d ),
∗
c∗t
∗
≔C(t, Xtπ , λ̂(t, IT , Yt ))
Xtπ
=
T −t+1
to be the optimal consumption-investment strategy. This strategy leads to the following wealth
process
∗
∗
1
dXtπ =Xtπ r + (σ λ̂(t, IT , Yt ) − r1d )⊤ Σ−1 (σ λ̂(t, IT , Yt ) − r1d ) −
dt
T −t+1
∗
+ Xtπ (σ λ̂(t, IT , Yt ) − r1d )⊤ σ −1 dWtIT Y ,
where the solution of this SDE is given by
 t

Z
∗
1
1
Xtπ =x0 exp  r + (σ λ̂(s, IT , Ys ) − r1d )⊤ Σ−1 (σ λ̂(s, IT , Ys ) − r1d ) −
ds
2
T −s+1
0
 t

Z
× exp  (σ λ̂(s, IT , Ys ) − r1d )⊤ σ −1 dWsIT Y  .
(5.29)
0
117
5 Continuous-Time Consumption-Investment Problems
In order to get an ansatz for the solution g of (5.28) we consider the expected reward under
strategy π ∗
Et(x,iT

 T
Z
∗
 log(c∗s )ds + log(XTπ ) | FtIT Y 
,y)
t


ZT
1
1
ds | FtIT Y 
r+ (σ λ̂(s, IT , Ys )−r1d )⊤ Σ−1 (σ λ̂(s, IT , Ys )−r1d )−
2
T −s+1
t
 T

Z
ZT
+ Et(x,iT ,y)  (σ λ̂(s, IT , Ys ) − r1d )⊤ σ −1 dWsIT Y + log(c∗s )ds | FtIT Y 
=Et(x,iT ,y) log(x)+
t
t
= log(x) + r(T − t) − log(T − t + 1)
 T
Z
1
+ Et(x,iT ,y) 
(σ λ̂(s, IT , Ys )−r1d )⊤ Σ−1 (σ λ̂(s, IT , Ys )−r1d ) + log
2
t
∗
Xsπ
T −s+1
!

ds | FtIT Y  .
For the last equality we use the martingale property of the Brownian integral. Now we plug in
the wealth process (5.29) and use also the martingale property
Et(x,iT
 T

Z
∗
I
Y
 log(c∗s )ds + log(XTπ ) | Ft T 
,y)
t
1
=(T − t + 1) log(x) + r(T − t) − log(T − t + 1) + r(T − t)2
2
ZT
ZT
+ log(T − s + 1) − log(T − t + 1)ds − log(T − s + 1)ds
t
t

 T
Z
1
(σ λ̂(s, IT , Ys ) − r1d )⊤ Σ−1 (σ λ̂(s, IT , Ys ) − r1d )ds | FtIT Y 
+ Et(x,iT ,y) 
2
t
 T s

Z Z
1
I
Y
+ Et(x,iT ,y) 
(σ λ̂(u, IT , Yu ) − r1d )⊤ Σ−1 (σ λ̂(u, IT , Yu ) − r1d )duds | Ft T 
2
t
t
1
=(T − t + 1) log(x) + r(T − t)( (T − t) + 1) − (T − t + 1) log(T − t + 1)
2

 T
Z
1
(σ λ̂(s, IT , Ys ) − r1d )⊤ Σ−1 (σ λ̂(s, IT , Ys ) − r1d )ds | FtIT Y 
+ Et(x,iT ,y) 
2
t
 T s

Z Z
1
+ Et(x,iT ,y) 
(σ λ̂(u, IT , Yu ) − r1d )⊤ Σ−1 (σ λ̂(u, IT , Yu ) − r1d )duds | FtIT Y  .
2
t
118
t
5.5 Solutions for Logarithmic and Power Utility Functions
We further simplify the expected reward under strategy π ∗ . Firstly, we consider for s ≥ t
h
i
Et(x,iT ,y) (σ λ̂(s, IT , Ys ) − r1d )⊤ Σ−1 (σ λ̂(s, IT , Ys ) − r1d ) | FtIT Y

⊤
Zs

=Et(x,iT ,y) σ −1 (σ λ̂(t, IT , Yt ) − r1d ) + ΣIuT dWuIT Y 
t

× σ −1 (σ λ̂(t, IT , Yt ) − r1d ) +
Zs
t


ΣIuT dWuIT Y  | FtIT Y 
=(σ −1 (σ λ̂(t, iT , y) − r1d ))⊤ (σ −1 (σ λ̂(t, iT , y) − r1d ))
 


⊤  t

⊤  s
Z
Zt
Z
Zs


+ E  ΣIuT dWuIT Y   ΣIuT dWuIT Y  | FtIT Y  −  ΣIuT dWuIT Y   ΣIuT dWuIT Y 
=(σ
⊤
0
0
0
0
−1
−1
(σ λ̂(t, iT , y) − r1d )) (σ (σ λ̂(t, iT , y) − r1d ))
 

⊤
Zs Zu
Zs 

+ Et(x,iT ,y)   ΣIrT dWrIT Y  ΣIuT dWuIT Y + tr ΣIuT ΣIuT du | FtIT Y 
0
0
0

⊤
Zt Zu
Zt I
I
Y
I
I
Y
T
T
T
T
− 2  Σr dWr  Σu dWu − tr ΣIuT ΣIuT du,
0
0
(5.30)
0
where for the last equality we apply Itô’s formula to f (Xt1 , . . . , Xtd ) := (Xt1 )2 + . . . + (Xtd )2 with
d
P
semimartingales dXtk :=
(ΣIt T )kj dWtIT Y j , k = 1, . . . , d, i.e.
j=1
⊤  s

 s

⊤
Z
Z
Zs Zu
Zs  ΣIuT WuIT Y   ΣIuT WuIT Y  = 2  ΣIrT dWrIT Y  ΣIuT dWuIT Y + tr ΣIuT ΣIuT du.
0
0
0
0
0
Using again the martingale property of the Brownian integral and properties of the trace (see
Lemma A.4.2) (5.30) simplifies to
h
i
Et(x,iT ,y) (σ λ̂(s, IT , Ys ) − r1d )⊤ Σ−1 (σ λ̂(s, IT , Ys ) − r1d ) | FtIT Y
=(σ
−1
=(σ
−1
⊤
−1
⊤
−1
(σ λ̂(t, iT , y) − r1d )) (σ
(σ λ̂(t, iT , y) − r1d )) (σ
(σ λ̂(t, iT , y) − r1d )) +
Zs
t
tr ΣIuT ΣIuT du
(σ λ̂(t, iT , y) − r1d )) − tr ΣIsT + tr ΣIt T .
119
5 Continuous-Time Consumption-Investment Problems
We then obtain with Fubini’s theorem
 T

Z
1
Et(x,iT ,y) 
(σ λ̂(s, IT , Ys ) − r1d )⊤ Σ−1 (σ λ̂(s, IT , Ys ) − r1d )ds | FtIT Y 
2
t

 T s
Z Z
1
I
Y
(σ λ̂(u, IT , Yu ) − r1d )⊤ Σ−1 (σ λ̂(u, IT , Yu ) − r1d )duds | Ft T 
+ Et(x,iT ,y) 
2
t
=
1
2
ZT
t
+
1
2
t
(σ λ̂(t, iT , y) − r1d )⊤ Σ−1 (σ λ̂(t, iT , y) − r1d ) − tr ΣIsT − ΣIt T ds
ZT Zs
t
t
(σ λ̂(t, iT , y) − r1d )⊤ Σ−1 (σ λ̂(t, iT , y) − r1d ) − tr ΣIuT − ΣIt T duds
1
1
= (T − t)( (T − t) + 1) (σ λ̂(t, iT , y) − r1d )⊤ Σ−1 (σ λ̂(t, iT , y) − r1d ) + tr ΣIt T
2
2
ZT Zs 1
IT
−
tr Σs + tr ΣIuT duds
2
t
t
d
X dI T
1
1
k
= (T − t)( (T − t) + 1) (σ λ̂(t, iT , y) − r1d )⊤ Σ−1 (σ λ̂(t, iT , y) − r1d ) +
IT
2
2
k=1 1 + dk t


ZT
d
1 X
−
log(1 + dIkT T ) − (T − t + 1) log(1 + dIkT t) + log(1 + dIkT s)ds .
2
k=1
!
t
For the last equality we again use the property of the trace as given in Lemma A.4.2. By
determining the integral in the above equation
ZT
t

I
I
log(1 + dIT T ) 1+dkT T − log(1 + dIT t) 1+dkT t − (T − t)
I
I
k
k
dkT
dkT
log(1 + dIkT s)ds =

0,
if dIkT , 0;
else
we obtain
 T

Z
1
Et(x,iT ,y) 
(σ λ̂(s, IT , Ys ) − r1d )⊤ Σ−1 (σ λ̂(s, IT , Ys ) − r1d )ds | FtIT Y 
2
t

 T s
Z Z
1
(σ λ̂(u, IT , Yu ) − r1d )⊤ Σ−1 (σ λ̂(u, IT , Yu ) − r1d )duds | FtIT Y 
+ Et(x,iT ,y) 
2
t
t
!
j
IT
X
d
1
1
k
= (T − t)( (T − t) + 1) (σ λ̂(t, iT , y) − r1d )⊤ Σ−1 (σ λ̂(t, iT , y) − r1d ) +
2
2
1
+
dIkT t
k=1
!
!
!
j
1 + dIkT t
1 + dIkT T
1X
IT
IT
+
log(1 + dk t) T − t + 1 +
− log(1 + dk T ) 1 +
+T −t .
2
dIT
dIT
k=1
120
k
k
5.5 Solutions for Logarithmic and Power Utility Functions
Hence, by summarizing all the calculations above we get

 T
Z
∗
Et(x,iT ,y)  log(c∗s )ds + log(XTπ ) | FtIT Y 
t
=(T − t + 1) log(x) − (T − t + 1) log(T − t + 1)
!
j
IT
X
d
1
1
1
k
+ (T − t)( (T − t) + 1) r + (σ λ̂(t, iT , y) − r1d )⊤ Σ−1 (σ λ̂(t, iT , y) − r1d ) +
2
2
2
1
+
dIkT t
k=1
!
!!
j
1 + dIkT t
1 + dIkT T
1X
1
IT
+
log(1 + dIkT t) T − t + 1 +
−
log(1
+
d
T
)
1
+
+ j(T − t).
k
IT
IT
2
2
d
d
k
k=1
k
Motivated by the above considerations we make the following ansatz for the solution of PDE
(5.28)
!
!
j
1 + dIkT t
1 + dIkT T
1
1X
log
− (T − t + 1) log(T − t + 1) + j(T − t)
1+
g(t, iT , y) = +
IT
IT
2
2
1 + dk T
dk
k=1
j
I
1
1 X dkT
+ (T − t)( (T − t) + 1) r +
2
2
1 + dI T t
k=1
k
!
1
⊤ −1
+ (σ λ̂(t, iT , y) − r1d ) Σ (σ λ̂(t, iT , y) − r1d ) .
2
Determining the relevant partial derivatives shows that g is indeed a solution to the PDE (5.28)
1
gt (t, iT , y) = log(T − t − 1) + 1 − j
2
j
I
1 X dkT
1
− (T − t + 1) r + (σ λ̂(t, iT , y) − r1d )⊤ Σ−1 (σ λ̂(t, iT , y) − r1d ) +
2
2
1 + dI T t
k=1
k
!
1
− (T − t)( (T − t) + 1)λ̂(t, iT , y)⊤ (ΣIt T )⊤ σ −1 (σ λ̂(t, iT , y) − r1d )
2
!2
!
j
j
I
X
dIkT
1 + dIkT T
1
1 X dkT
1
+
1+
− (T − t)( (T − t) + 1)
IT
IT
2
2
2
1
+
d
t
1
+
d
t
dIkT
k
k
k=1
k=1
1
= log(T − t + 1) + 1 − (T − t + 1)(r + (σ λ̂(t, iT , y) − r1d )⊤ Σ−1 (σ λ̂(t, iT , y) − r1d ))
2
1
⊤
− (T − t)( (T − t) + 1)λ̂(t, iT , y) (ΣIt T )⊤ σ −1 (σ λ̂(t, iT , y) − r1d )
2
1
1
− (T − t)( (T − t) + 1)tr ΣIt T ΣIt T
2
2
1
gy (t, iT , y) =(T − t)( (T − t) + 1)(ΣIt T )⊤ σ −1 (σ λ̂(t, iT , y) − r1d )
2
1
gyy (t, iT , y) =(T − t)( (T − t) + 1)ΣIt T ΣIt T .
2
Theorem 5.5.2. Let Assumption 5.2.20 be satisfied. Furthermore, let without loss of generality
T
dIj+1
= . . . = dIdT = 0 for some j ∈ {0, . . . , d}. Then it holds:
121
5 Continuous-Time Consumption-Investment Problems
a) The value function is given by
V (t, x, iT , y) = (T − t + 1) log(x) + g(t, iT , y),
where
j
1X
1
log
g(t, iT , y) := − (T − t + 1) log(T − t + 1) + j(T − t) +
2
2
k=1
1 + dIkT t
1 + dIkT T
!
1+
1 + dIkT T
dIkT
!
j
I
1 X dkT
1
+ (T − t)( (T − t) + 1) r +
2
2
1 + dIT t
k=1
k
!
1
⊤ −1
+ (σ λ̂(t, iT , y) − r1d ) Σ (σ λ̂(t, iT , y) − r1d ) .
2
∗
b) The optimal investment strategy is given by αt∗ = Π(t, Xtπ , IT , Yt ), where
Π(t, x, iT , y) := Σ−1 (σ λ̂(t, iT , y) − r1d ).
∗
The optimal consumption strategy is given by c∗t = C(t, Xtπ , IT , Yt ), where
C(t, x, iT , y) :=
x
.
T −t+1
As in the discrete-time consumption-investment problem, the value function can be separated
into two parts: The first part depends on the wealth and the second part depends on the insider
information and the observation. Moreover, the certainty equivalence principle holds: Compared
to the classical Merton proportion we only have to replace the constant drift λ by the estimator of
the insider drift λ̂(t, IT , Yt ) in order to get the optimal investment strategy. Whereas the optimal
investment strategy depends explicitly on the insider information and on the observation, the
optimal consumed fraction of wealth is only a deterministic time-dependent function.
Proof. Since the solution v(t, x, iT , y) of the HJB equation given in Theorem 5.5.1 is a function
in C 1,2,2,2 ([0, T ], R>0 , Rd , Rd ) it follows by verification that the value function of (P c ) is given
by V (t, x, iT , y) = v(t, x, iT , y). Furthermore, the candidate optimal strategy π ∗ = (c∗t , αt∗ )t∈[0,T ]
which we get from the maximization problem within the HJB equation (see (5.25) and (5.24)) is
indeed the optimal strategy (for the verification theorem see e.g. Pham [2009] Theorem 3.5.2). Remark 5.5.3. In the one-dimensional case (d = 1) the value function simplifies to
V (t, x, iT , y) = (T − t + 1) log(x) + g(t, iT , y),
where

1
1
1 ΣIaT
+
g(t, iT , y) =(T − t)( (T − t) + 1) r +
I
T
2
2 1 + Σa t 2
σ λ̂(t, iT , y) − r
σ
1
− (T − t + 1) log(T − t + 1) + log(1 + ΣIaT t)(T − t + 1)
2
ZT
1
1
− log(1 + ΣIaT T ) −
log(1 + ΣIaT s)ds
2
2
t
122
!2 

5.5 Solutions for Logarithmic and Power Utility Functions
with
ZT
t

I
I
log(1 + ΣIT T ) 1+ΣaT T − log(1 + ΣIT t) 1+ΣaT t − (T − t)
a
a
IT
IT
IT
Σa
Σa
log(1 + Σa s)ds =
0,
if ΣIaT , 0;
else.
The optimal investment strategy is given by αt∗ = Π(t, Xtπ∗ , IT , Yt ), where
Π(t, x, iT , y) =
σ λ̂(t, iT , y) − r
.
σ2
In the one-dimensional case the behavior of the optimal investment strategy towards changes in
the realized values of IT and Yt is similar to the behavior of the h-investor’s optimal strategy
(compare Corollary 4.6.25 and Corollary 4.6.26). We see that the optimal investment strategy
is strictly and linearly increasing in the realized value iT of the insider information (compare
Remark 5.5.3). Whether or not the optimal strategy is increasing in the realized value y of Yt
depends on the sign of ΣIaT . Contrary to the h-investor’s value function which is increasing in iT
and y for ΣIaT > 0, the continuous-time value function depends quadratically on iT and y. The
main reason for the different behavior of the value functions is that short selling is not optimal
for the h-investor. When the insider information is "bad" (e.g. iT ≪ 0) the h-insider does not
invest into the stock at all, whereas the continuous-time investor short sells the stock such that
he can buy additional bonds.
For the terminal wealth problem we obtain the following result:
Theorem 5.5.4. Let Assumption 5.2.20 be satisfied. For the terminal wealth problem it holds:
a) The value function is given by
V (t, x, iT , y) = (T − t + 1) log(x) + g(t, iT , y),
where
d
I
1
1 X dkT
g(t, iT , y) = (T − t) r + (σ λ̂(t, iT , y) − r1d )⊤ Σ−1 (σ λ̂(t, iT , y) − r1d ) +
IT
2
2
k=1 1 + dk t
!
d
1 + dIkT t
1X
.
+
log
2
1 + dIkT T
k=1
!
∗
b) The optimal investment strategy is given by αt∗ = Π(t, Xtπ , IT , Yt ), where
Π(t, x, iT , y) := Σ−1 (σ λ̂(t, iT , yt ) − r1d ).
The optimal investment strategy is the same as in the consumption-investment problem (compare
Theorem 5.5.2).
5.5.2 Power Utility Functions
Here we assume Uc (x) = Up (x) = xγ , 0 < γ < 1.
123
5 Continuous-Time Consumption-Investment Problems
Theorem 5.5.5. Let Assumption 5.2.20 be satisfied. Furthermore, let γ(1 + dIkT T ) < 1 for all
k = 1, . . . , d. Then the function
v(t, x, i, y) := g(t, i, y)1−γ xγ
solves the HJB equation, where
g(t, iT , y) :=
ZT
ef (t,iT ,y,s) ds + ef (t,iT ,y,T )
(5.31)
and
t

!−1
d
dIkT
γ
1X
γ

log
1+
r(s − t) +
f (t, iT , y, s) := −
(s − t)
γ −1
2
γ − 1 1 + dIT t
k
k=1
+
1 + dIkT s
1 + dIkT t
γ
1
(s − t)(σ λ̂(t, iT , y) − r1d )⊤ σ −1
2 (γ − 1)2
−1 −1
γ
σ (σ λ̂(t, iT , y) − r1d ).
(s − t)ΣIt T
× Id +
γ −1
γ 
!− 1−γ

Proof. By solving the maximization problem within the HJB equation we get the candidate
(optimal) consumption strategy
C(t, x, i, y) =
vx (t, x, i, y)
γ
1
γ−1
(5.32)
and the candidate (optimal) investment strategy
Π(t, x, iT , y) = −Σ−1
vx (t, x, iT , y)(σ λ̂(t, iT , y) − r1d ) + σvxy (t, x, iT , y)
xvxx (t, x, iT , y)
(5.33)
Plugging in the candidate (optimal) strategy into the HJB equation gives the following second
order non-linear PDE
γ
γ
1
0 =vt (t, x, iT , y) + (vx (t, x, iT , y)) γ−1 (γ − γ−1 − γ − γ−1 ) + rxvx (t, x, iT , y)
vx (t, x, iT , y) vx (t, x, iT , y)(σ λ̂(t, iT , y) − r1d )⊤
−
vxx (t, x, iT , y)
+ vxy (t, x, iT , y)⊤ σ Σ−1 (σ λ̂(t, iT , y) − r1d ) + vy (t, x, iT , y)⊤ λ̂(t, iT , y)
1
1
vx (t, x, iT , y)(σ λ̂(t, iT , y) − r1d )⊤ + vxy (t, x, iT , y)⊤ σ Σ−1 ·
+
2 vxx (t, x, iT , y)
· vx (t, x, iT , y)(σ λ̂(t, iT , y) − r1d ) + σvxy (t, x, iT , y)
1
−
vx (t, x, iT , y)(σ λ̂(t, iT , y) − r1d )⊤ + vxy (t, x, iT , y)⊤ σ σ −1 vxy (t, x, iT , y)
vxx (t, x, iT , y)
1
(5.34)
+ tr (vyy (t, x, iT , y)) .
2
Motivated by the h-investor’s value function which can be separated into a product consisting of
the utility of the wealth and a function depending on the insider information and the observation,
we are making the ansatz
v(t, x, iT , y) = g(t, iT , y)1−γ xγ ,
124
5.5 Solutions for Logarithmic and Power Utility Functions
where g : [0, T ] × Rd × Rd → R satisfies the terminal condition g(T, iT , y) = 1. The partial derivatives needed in the HJB equation are given by
vt (t, x, iT , y) = − (γ − 1)xγ g(t, iT , y)−γ gt (t, iT , y);
vx (t, x, iT , y) =γxγ−1 g(t, iT , y)1−γ ;
vxx (t, x, iT , y) =γ(γ − 1)xγ−2 g(t, iT , y)1−γ ;
vy (t, x, iT , y) = − (γ − 1)xγ g(t, iT , y)−γ gy (t, iT , y);
vxy (t, x, iT , y) = − γ(γ − 1)xγ−1 g(t, iT , y)−γ gy (t, iT , y);
vyy (t, x, iT , y) =γ(γ − 1)xγ g(t, iT , y)−γ−1 gy (t, iT , y)(gy (t, iT , y))⊤
− (γ − 1)xγ g(t, iT , y)−γ gyy (t, iT , y).
Substituting these partial derivatives into (5.34) gives
1
0 =1 + gt (t, iT , y) + tr(gyy (t, iT , y))
2
1
γ
γ
⊤ −1
+ g(t, iT , y)
(σ
λ̂(t,
i
,
y)
−
r1
)
Σ
(σ
λ̂(t,
i
,
y)
−
r1
)
−
r
T
T
d
d
2 (γ − 1)2
γ −1
γ
σ −1 (σ λ̂(t, iT , y) − r1d ) − λ̂(t, iT , y)).
(5.35)
− gy (t, iT , y)⊤ (
γ −1
Firstly, we consider the homogeneous part of this PDE
1
0 =g̃t (t, iT , y) + tr(g̃yy (t, iT , y))
2
1
γ
γ
⊤ −1
+ g̃(t, iT , y)
(σ
λ̂(t,
i
,
y)
−
r1
)
Σ
(σ
λ̂(t,
i
,
y)
−
r1
)
−
r
T
T
d
d
2 (γ − 1)2
γ −1
γ
σ −1 (σ λ̂(t, iT , y) − r1d ) − λ̂(t, iT , y)),
(5.36)
− g̃y (t, iT , y)⊤ (
γ −1
where g̃(T, iT , y) = 1. We remark here that (5.36) is the PDE which has to be solved in the
terminal wealth problem. We make the ansatz
g̃(t, iT , y) = ef (t,iT ,y)
with terminal condition f (T, iT , y) = 0. The relevant partial derivatives are given by
g̃t (t, iT , y) =g̃(t, iT , y)ft (t, iT , y);
g̃y (t, iT , y) =g̃(t, iT , y)fy (t, iT , y);
g̃yy (t, iT , y) =g̃(t, iT , y)fyy (t, iT , y) + g̃(t, iT , y)fy (t, iT , y, s)fy (t, iT , y)⊤
such that the PDE (5.36) reduces to
1
1
0 =ft (t, iT , y) + tr(fyy (t, iT , y)) + fy (t, iT , y)⊤ fy (t, iT , y)
2
2
γ
⊤
−1
− fy (t, iT , y) (
σ (σ λ̂(t, iT , y) − r1d ) − λ̂(t, iT , y))
γ −1
γ
γ
1
(σ λ̂(t, iT , y) − r1d )⊤ Σ−1 (σ λ̂(t, iT , y) − r1d ) −
r.
+
2
2 (γ − 1)
γ −1
(5.37)
We make a further ansatz
f (t, iT , y) =
1 (1)
1 1
k (t) +
(σ λ̂(t, iT , y) − r1d )⊤ σ −1 K (2) (t)σ −1 (σ λ̂(t, iT , y) − r1d )
γ −1
2 γ −1
125
5 Continuous-Time Consumption-Investment Problems
with functions k (1) : [0, T ] → R and K (2) : [0, T ] → Rd×d (symmetric) with terminal conditions
(2)
(1)
d
d (1)
k (t) and Kt (t) := dt
K (2) (t). Then, with
k (1) (T ) = 0 and K (2) (T ) = 0d×d . Let kt (t) := dt
the partial derivatives
1 (1)
1
k (t) −
λ̂(t, iT , y)⊤ (ΣIt T )⊤ K (2) (t)σ −1 (σ λ̂(t, iT , y) − r1d )
γ −1 t
γ −1
1 1
(2)
(σ λ̂(t, iT , y) − r1d )⊤ σ −1 Kt (t)σ −1 (σ λ̂(t, iT , y) − r1d );
+
2 γ −1
1
ΣIT K (2) (t)σ −1 (σ λ̂(t, iT , y) − r1d );
fy (t, iT , y) =
γ −1 t
1
fyy (t, iT , y) =
ΣIT K (2) (t)ΣIt T ,
γ −1 t
ft (t, iT , y) =
we get
0=
γ
1 1
1 (1)
kt (t) −
r+
tr ΣIt T K (2) (t)ΣIt T
γ −1
γ −1
2 γ −1
1 1
1
1
(2)
+ (σ λ̂(t, iT , y) − r1d )⊤
σ −1 Kt (t)σ −1 +
σ −1 K (2) (t)ΣIt T ΣIt T K (2) (t)σ −1
2 γ −1
2 (γ − 1)2
γ
1
γ
IT −1
−1 (2)
−1
−
σ K (t)Σt σ +
Σ
(σ λ̂(t, iT , y) − r1d ).
(γ − 1)2
2 (γ − 1)2
Since the equation has to hold for all iT , y ∈ Rd it follows that we have to solve the following two
ODEs
(I) The Riccati-type (matrix) ODE
(2)
Kt (t)
= −
γ
γ
1
K (2) (t)ΣIt T ΣIt T K (2) (t) + 2
K (2) (t)ΣIt T −
Id ;
γ −1
γ −1
γ −1
(II) The ODE
1
(1)
kt (t) = γr − tr ΣIt T K (2) (t)ΣIt T .
2
Determining the derivative of
K (2) (t) :=
−1
γ
γ
(T − t) Id +
(T − t)ΣIt T
γ −1
γ −1
shows that K (2) (t) solves indeed the Riccati-type ODE with terminal condition K (2) (T ) = 0.
Using K (2) (t) we are able to solve the second ODE (II) with terminal condition k (1) (T ) = 0. By
Lemma A.4.2 we get
I
tr(ΣIt T K (2) (t)ΣIt T ) =
d
X
k=1
126
γ(T − t)
(dkT )2
I
(1+dkT t)2
I
γ
1+dkT T
I
1+dkT t
−1
.
5.5 Solutions for Logarithmic and Power Utility Functions
Hence,
k
(1)
1
(t) =γrt −
2
Zt
tr(ΣIsT K (2) (s)ΣIsT )ds + c
0
I
=γrt +
1
2
d Zt
X
k=1 0
γ(T − s)
=γrt +
1
2
k=1 0
I
(1+dkT s)2
I
1−γ
1+dkT T
ds + c
I
1+dkT s
I
d Zt
X
(dkT )2
I
I
γdkT (1+dkT T )
I
(1+dkT
s)2
−
γdkT
I
1+dkT s
I
1−γ
1+dkT T
ds + c
I
1+dkT s
d
1 + dIkT T
1X
=γrt +
log 1 − γ
2
1 + dI T s
k
k=1
!
t
t
I
I
− γ log((1 + dkT s) − γ(1 + dkT T ))
0
0
!
+c
(5.38)
I
for a constant c ∈ R. Note that due to the assumptions we have that γ
1+dkT T
I
1+dkT s
< 1 for all s ∈ [0, T ].
Thus, the logarithm in (5.38) is well-defined. With the terminal condition k (1) (T ) = 0 it follows
that

!1−γ
!γ 
d
IT
IT
X
d
1
+
d
T
1
γ
k
k
.
k (1) (t) = −γr(T − t) +
log  1 +
(T − t)
2
γ − 1 1 + dIT t
1 + dI T t
k
k=1
k
Hence, g̃(t, iT , y) = ef (t,iT ,y) solves the homogeneous PDE (5.36). We now claim that the solution
of the inhomogeneous PDE (5.35) with terminal condition g(T, iT , y) = 1 is given by
g(t, iT , y) =
ZT
ef (t,iT ,y,s) ds + ef (t,iT ,y,T ) ,
t
where

!−1
d
dIkT
1X
γ
γ

r(s − t) +
log
1+
(s − t)
f (t, iT , y, s) := −
γ −1
2
γ − 1 1 + dIT t
k=1
+
k
1 + dIkT s
1 + dIkT t
1
γ
(s − t)(σ λ̂(t, iT , y) − r1d )⊤ σ −1
2 (γ − 1)2
−1 −1
γ
(s − t)ΣIt T
σ (σ λ̂(t, iT , y) − r1d ).
× Id +
γ −1
γ 
!− 1−γ

Note that f (t, iT , y, T ) = f (t, iT , y). Using Leibniz’s rule for differentiation under an integral (parameter dependent integrals) it can easily be seen that g solves indeed (5.35). The idea of
constructing a solution of the inhomogeneous PDE from the solution of the corresponding homogeneous one is known as Duhamel’s principle (see for instance Evans [2010]).
Theorem 5.5.6. Let the assumptions of Theorem 5.5.5 be satisfied. Then it holds:
a) The value function is given by
V (t, x, iT , y) = xγ g(t, iT , y)1−γ ,
127
5 Continuous-Time Consumption-Investment Problems
where
g(t, iT , y) :=
ZT
ef (t,iT ,y,s) ds + ef (t,iT ,y,T )
(5.39)
and
t
γ
r(s − t)
γ −1

!−1
d
X
dIkT
1
γ
+
log  1 +
(s − t)
2
γ − 1 1 + dIT t
f (t, iT , y, s) := −
k
k=1
1 + dIkT s
1 + dIkT t
γ 
!− 1−γ

1
γ
+
(s − t)(σ λ̂(t, iT , y) − r1d )⊤ σ −1
2 (γ − 1)2
−1 −1
γ
(s − t)ΣIt T
σ (σ λ̂(t, iT , y) − r1d ).
× Id +
γ −1
∗
b) The optimal investment strategy is given by αt∗ = Π(t, Xtπ , IT , Yt ), where
Π(t, x, iT , y) :=
1
Σ−1 (σ λ̂(t, iT , y) − r1d )
1−γ
RT f (t,i ,y,s)
T
e
fy (t, iT , y, s)ds + ef (t,iT ,y,T ) fy (t, iT , y, T )
−1 t
+σ
.
RT
f
(t,i
,y,s)
f
(t,i
,y,T
)
T
T
e
ds + e
t
∗
The optimal consumption strategy is given by c∗t = C(t, Xtπ , IT , Yt ), where
C(t, x, iT , y) := T
R
x
.
ef (t,iT ,y,s) ds + ef (t,iT ,y,T )
t
The value function can be written as the product of the power utility of the wealth and a function
which depends on the insider information and the observation. The optimal investment strategy
and the optimal consumed fraction of wealth depend on the insider information and the observation, but do not depend on the level of wealth. Moreover, we can write the optimal investment
strategy as a sum consisting of the (modified) Merton proportion, where the constant drift λ is
replaced by the estimator λ̂(t, IT , Yt ), and an additional hedging part which is often called the
drift risk. Note that in the classical Merton problem without partial and insider information
the drift risk disappears.
Proof. Since the solution v(t, x, iT , y) of the HJB equation given in Theorem 5.5.5 is a function in
C 1,2,2,2 ([0, T ], R≥0 , Rd , Rd ) it follows analogously to Theorem 5.5.2 by verification that the value
function of the optimization problem (P c ) is given by V (t, x, iT , y) = v(t, x, iT , y). The candidate
optimal strategy π ∗ = (c∗t , αt∗ ) which we get from the maximization problem within the HJB
equation (see (5.32) and (5.33)) is indeed the optimal strategy of (P c ).
Remark 5.5.7. In the one-dimensional case (d = 1) the value function simplifies to
V (t, x, iT , y) = xγ g(t, iT , y)1−γ ,
128
5.5 Solutions for Logarithmic and Power Utility Functions
where
g(t, iT , y) =
ZT
f (t,iT ,y,T )
ef (t,iT ,y,s)ds+e
and
t
γ
1
γ(T − t)
f (t, iT , y, s) =
r(T − t) +
γ −1
2 (1 − γ)2 (1 − γ ΣIt T (T − t))
1−γ
γ !
IT
(1 + Σt (T − t)) γ−1
1
.
+ log
2
1 + γ ΣIt T (T − t)
σ λ̂(t, iT , y) − r
σ
!2
γ−1
The optimal investment strategy is given by αt∗ = Π(t, Xtπ∗ , IT , Yt ), where
σ λ̂(t, iT , y) − r
Π(t, x, iT , y) =
+
(1 − γ)σ 2
RT
ef (t,iT ,y,s) fy (t, iT , y, s)ds + ef (t,iT ,y,T ) fy (t, iT , y, T )
t
RT
σ ef (t,iT ,y,s) ds + ef (t,iT ,y,T )
t
∗
and the optimal consumption strategy is given by c∗t = C(t, Xtπ , IT , Yt ), where
C(t, x, iT , y) := T
R
x
.
ef (t,iT ,y,s) ds + ef (t,iT ,y,T )
t
Remark 5.5.8. Due to property (2.1) there is an interesting connection between the optimization
problems with logarithmic and power utility functions. More precisely, the (modified) value and
the optimal strategy of the consumption-investment problem with power utility functions converge
to the value and the optimal strategy with logarithmic utility functions as γ tends to zero: For this
purpose, we write f (t, iT , y, s, γ) and g(t, iT , y, γ), respectively, for the functions f and g given in
(5.39). Firstly, we consider the (modified) value
 T

Z
xγ g(0, iT , 0, γ)1−γ − (T + 1)
1
I
Y
sup E  (cγs − 1)ds + ((XTπ )γ − 1) | F0 T  = 0
.
γ π∈A0
γ
0
Using dominated convergence it is straightforward to show that lim g(0, iT , 0, γ) = T + 1. Then we
γ→0
obtain by l’Hôspital’s rule
xγ0 g(0, iT , 0, γ)1−γ − (T + 1)
γ→0
γ
∂
= lim xγ0 log(x0 )g(0, iT , 0, γ)1−γ + (1 − γ)g(0, iT , 0, γ)−γ g(0, iT , 0, γ)
γ→0
∂γ
− g(0, iT , 0, γ)1−γ log(g(0, iT , 0, γ) .
lim
T
Let dIj+1
= . . . = dIdT = 0 for some j ∈ {0, . . . , d} (w.l.o.g). By using again dominated convergence
129
5 Continuous-Time Consumption-Investment Problems
we get
∂
g(0, iT , 0, γ)
∂γ

 T
Z
∂
∂
= lim  ef (0,iT ,0,s,γ) f (0, iT , 0, s, γ)ds + ef (0,iT ,0,T,γ) f (0, iT , 0, T, γ)
γ→0
∂γ
∂γ
lim
γ→0
0
!
j
1
1 X IT 1
=T ( T + 1) r +
dk + (σ λ̂(0, iT , 0) − r1d )⊤ Σ−1 (σ λ̂(0, iT , 0) − r1d )
2
2
2
k=1
!
j
1 + dIkT T
1
1X
IT
log(1 + dk T )
+1 + Tj
−
IT
2
2
d
k
k=1
since
∂
f (0, iT , 0, s, γ)
∂γ
d
1
1X
γ IT
γ IT −2 IT
1
rs
+
(1 +
d s)(1 +
d s) dk s
=
(γ − 1)2
2
(γ − 1)2
γ −1 k
γ −1 k
k=1
−
and
d
1X
2
k=1
γ IT −1
γ IT
1
(1 +
d s)(1 +
d s) log(1 + dIkT s)
(γ − 1)2
γ −1 k
γ −1 k
γ
1 γ +1
s(σ λ̂(0, iT , 0) − r1d )⊤ σ −1 (Id +
sΣIT )−1 σ −1 (σ λ̂(0, iT , 0) − r1d )
−
3
2 (γ − 1)
γ −1 a
−1 I
γ
γ
1
ΣaT
s2 (σ λ̂(0, iT , 0) − r1d )⊤ σ −1 Id +
sΣIT
+
2 (γ − 1)4
γ −1 a
−1 −1
γ
× Id +
sΣIT
σ (σ λ̂(0, iT , 0) − r1d ).
γ −1 a
lim
γ→0
∂
f (0, iT , 0, s, γ)
∂γ
d
=rs +
1
1 X IT
(dk s − log(1 + dIkT s)) + s(σ λ̂(0, iT , 0) − r1d )⊤ Σ−1 (σ λ̂(0, iT , 0) − r1d ).
2
2
k=1
Altogether, we have
 T

Z
1
I
Y
lim
sup E  (cγs − 1)ds + ((XTπ )γ − 1) | F0 T 
γ→0 γ π∈A0
0
!
!
j
1 + dIkT T
1
1
1X
=(T + 1) log(x0 ) − (T + 1) log(T + 1) + jT +
log
1+
2
2
1 + dIkT T
dIkT
k=1
#
"
j
⊤ −1
1 X IT 1
1
σ λ̂(0, iT , 0) − r1d
dk + σ λ̂(0, iT , 0) − r1d Σ
+ T ( T + 1) r +
2
2
2
k=1
 T

Z
= sup E  log(cs )ds + log(XTπ ) | F0IT Y  .
π∈A0
130
0
5.6 Properties of the Continuous-Time Problems
Analogously, by dominated convergence it follows that the optimal consumption-investment strategy given in Theorem 5.5.6 converges for γ → 0 to the optimal strategy given in Theorem 5.5.2.
The terminal wealth problem can be solved analogously. As mentioned in the proof of Theorem
5.5.5 one has to solve the homogeneous PDE (5.36) in order to get a solution of the associated
HJB equation.
Theorem 5.5.9. Let the assumptions of Theorem 5.5.5 be satisfied. For the terminal wealth
problem it holds:
a) The value function is given by
V (t, x, iT , y) = xγ e(1−γ)f (t,iT ,y) ,
where
γ
r(T − t)
γ −1

!−1
d
dIkT
1X
γ
(T − t)
+
log  1 +
2
γ − 1 1 + dI T t
f (t, iT , y) := −
k
k=1
1 + dIkT T
1 + dIkT t
γ 
!− 1−γ
1
γ
+
(T − t)(σ λ̂(t, iT , y) − r1d )⊤ σ −1
2 (γ − 1)2
−1 −1
γ
(T − t)ΣIt T
σ (σ λ̂(t, iT , y) − r1d ).
× Id +
γ −1

∗
b) The optimal investment strategy is given by αt∗ = Π(t, Xtπ , iT , Yt ), where
1
Σ−1 (σ λ̂(t, iT , y) − r1d ) + σ −1 fy (t, iT , y)
1−γ
1
γ
=
σ −1 (Id +
(T − t)(Id + ΣIaT t)−1 ΣIaT )−1 σ −1 (σ λ̂(t, iT , y) − r1d ).
1−γ
γ −1
Π(t, x, iT , y) =
5.6 Properties of the Continuous-Time Problems
5.6.1 The Value of Information
As for the discrete-time consumption-investment problems, we define the value of the insider
information IT as the certainty equivalent (CEIT ) - also called indifference price (studied also in
Amendinger et al. [1998, 2003], Hansen [2013] and Liu et al. [2010]). By setting the convex factor
to a = 0 we obtain the value function of the regular investor. The latter is denoted by
V a=0 (t, x, y).
The value function V (t, x, iT , y) is different for the different types of insider information. If it
is important which type of insider information we consider we write V X (t, x, iT , y) for the value
function with insider information ITX , where X ∈ {S, W, λ} stands for stock price, Brownian or
drift information.
131
5 Continuous-Time Consumption-Investment Problems
Definition 5.6.1. The certainty equivalent of the insider information IT is defined as the solution
CEIT of the following equation
V a=0 (0, x0 + CEIT , 0) =
Z
V (0, x0 , iT , 0)QI0T (diT ),
where QI0T denotes the distribution of the insider information IT .
Definition 5.6.2. The certainty equivalent between the insider information ITX and the insider
information ITY is defined as the solution CEXY of the following equation
Z
Z
X
X
V (0, x0 , iT + CEXY , 0)Q0 (diT ) = V Y (0, x0 , iT , 0)QY0 (diT ).
The variables X, Y stand for stock price, Brownian or drift information.
X
′
We denote by ΣX
a′ the modified covariance matrix associated to IT with convex factor a and by
Y
Y
Σa the modified covariance matrix associated to IT with convex factor a. Furthermore, we write
Y
d0k , k = 1, . . . , d, for the eigenvalues of Σ0 and dX
k , dk for the respective eigenvalues of the modified
covariance matrices.
Logarithmic Utility Functions
Here we assume Uc (x) = Up (x) = log(x). Hence, for a state (x, iT , y) the value function is given
by
V (t, x, iT , y) = (T − t + 1) log(x) + g(t, iT , y),
with g(t, iT , y) given in Theorem 5.5.2. The function g is different for the three types of insider
information. Thus, we write g X , X ∈ {S, W, λ}, for insider information ITX . If the investor does
not have insider information, then we write g a=0 (t, y).
T
= . . . = dIdT = 0 for some j ∈ {0, . . . , d}.
Theorem 5.6.3.
a) Without loss of generality, let dIj+1
Then the certainty equivalent is given by
"
CEIT =x0 exp
!
j
X
1 + dIkT (T + 1) 1
IT
T (j − d) −
log(1 + dk T )
2(T + 1)
dIkT
k=1
!
#
d
X
1 + d0k (T + 1)
1
0
× exp
−1 ,
log(1 + dk T )
2(T + 1)
d0k
k=1
X
Y
Y
b) Without loss of generality, let dX
j+1 = . . . = dd = dl+1 = . . . = dd = 0 for some j, l ∈ {0, . . . , d}.
Then the certainty equivalent between the insider information ITX and ITY is given by
"
CEXY =x0 exp
l
X
1 + dYk (T + 1) 1
T (l − j) −
log(1 + dYk T )
2(T + 1)
dYk
k=1
j
× exp
132
!
X
1 + dX
1
k (T + 1)
log(1 + dX
k T)
2(T + 1)
dX
k
k=1
!
#
−1 .
5.6 Properties of the Continuous-Time Problems
Proof.
a) It follows directly from Theorem 5.5.2 that
E[g(0, IT , 0)] − g a=0 (0, 0)
−1 .
CEIT = x0 exp
T +1
By the explicit representation of the function g (see Theorem 5.5.2) and using the following
relation
E[(σ λ̂(0, IT , 0) − r1d )⊤ Σ−1 (σ λ̂(0, IT , 0) − r1d )]
=(σλ0 − r1d )⊤ Σ−1 (σλ0 − r1d ) − tr(ΣIaT − Σ0 )
=(σλ0 − r1d )⊤ Σ−1 (σλ0 − r1d ) −
j
X
k=1
dIkT +
d
X
d0k
(5.40)
k=1
we obtain the assertion.
b) Due to Theorem 5.5.2 the certainty equivalent between the insider information ITX and ITY
is given by
E[g Y (0, ITY , 0)] − E[g X (0, ITX , 0)]
−1 .
CEXY = x0 exp
T +1
Analogously to part a), the assertion follows using (5.40).
Y
Theorem 5.6.4. If ΣX
a′ = Σa , then the investor is indifferent between the insider information
X
Y
IT and IT (i.e. CEXY = 0).
Y
X
Y
Proof. Due to ΣX
a′ = Σa it holds that dk = dk , k = 1, . . . , d. Consequently, the statement follows
from Theorem 5.6.3.
Theorem 5.6.5. Let Σ0 and Σε be simultaneously diagonalizable. Then stock price information
is more valuable than Brownian information (with the same convex factor a ∈ [0, 1)). Moreover,
for T ≥ 1 stock price information is more valuable than drift information (with the same convex
factor a ∈ [0, 1)).
Proof. Let d0k , dεk , k = 1, . . . , d, denote the eigenvalues of Σ0 and Σε . Since by assumption Σ0
and Σε are simultaneously diagonalizable, it is straightforward to determine for k = 1, . . . , d the
W λ
S
W
λ
respective eigenvalues dS
k , dk , dk of Σa , Σa and Σa , namely
dS
k =
(1 − a)4 (dεk )2 d0k
−a2
;
+
a2 T + (1 − a)2 dεk (a2 T + (1 − a)2 dεk )(a2 T 2 d0k + a2 T + (1 − a)2 dεk )
0
dW
k =dk −
dλ
k =
a2 T
a2
;
+ (1 − a)2 dεk
(1 − a)2 dεk d0k
a2 d0k + (1 − a)2 dεk
λ
S
W
Then for all a ∈ [0, 1) it holds that dS
k ≤ dk for T ≥ 1 and dk ≤ dk . Moreover, the function
(
+1)
, x , 0, x > − T1
log(1 + xT ) 1+x(T
x
f (x) :=
T,
x=0
133
5 Continuous-Time Consumption-Investment Problems
is non-decreasing as
f ′ (x) =
(
1+xT
T
1+xT (1 + x
T ( 21 T + 1),
) − x12 log(1 + xT ), x , 0
x=0
2
x T
with f ′ (x) ≥ 0 if and only if 1+xT
+ xT ≥ log(1 + xT ), x , 0. The latter inequality holds for all
1
x > − T due to the estimation
log(1 + xT ) ≤ xT ≤ xT +
x2 T
.
1 + xT
| {z }
≥0
W
S
λ
Since dS
k ≤ dk and dk ≤ dk (for T ≥ 1) the assertion follows from Theorem 5.6.3 part b) using
the monotonicity of f .
Remark 5.6.6. In the one-dimensional case (d = 1) stock price information is always more
valuable than Brownian information (with the same convex factor a ∈ [0, 1)).
Power Utility Functions
Now we assume Uc (x) = Up (x) = xγ , 0 < γ < 1. Let the assumptions of Theorem 5.5.6 be satisfied.
Then for a state (x, iT , y) the value function is given by
V (t, x, iT , y) = xγ g(t, iT , y)1−γ
(for the function g see Theorem 5.5.6). The function g is again different for the different types
of insider information. Thus, we also write here g X for the insider information ITX and we write
g a=0 (t, y) if the investor does not have additional information. It follows directly from Theorem
5.5.6:
Theorem 5.6.7.
a) The certainty equivalent is given by
#
"
1
E[g(0, IT , 0)1−γ ] γ
−1 .
CEIT = x0
g a=0 (0, 0)1−γ
b) The certainty equivalent between the insider information ITX and ITY is given by
"
#
1
E[g Y (0, ITY , 0)1−γ ] γ
CEXY = x0
−1 .
E[g X (0, ITX , 0)1−γ ]
Here, contrary to Theorem 5.6.3, we cannot simplify the representation of the certainty equivalent
considerably. However, for the terminal wealth problem (without consumption) we are able to
present CEIT and CEXY in an explicit form by virtue of the normal distribution of λ̂(0, IT , 0).
Corollary 5.6.8. Let the assumptions of Theorem 5.5.9 be satisfied. Then for the terminal wealth
problem it holds:
a) The certainty equivalent is given by

d
Y
CEIT = x0 
k=1
134
γ
(1 + γ−1
dIkT T )(1 + d0k T )
(1 +
IT
γ
0
γ−1 dk T )(1 + dk T )
! 21


− 1 .
5.6 Properties of the Continuous-Time Problems
b) The certainty equivalent between the insider information ITX and ITY is given by

CEXY = x0 
d
Y
γ
dYk T
1 + γ−1
1 + dYk T
k=1
1 + dX
k T
γ
1 + γ−1
dX
k T
! 21

− 1 .
c) If additionally Σ0 and Σε are simultaneously diagonalizable, then CEW S ≥ 0 and CEλS ≥ 0
for T ≥ 1.
Proof. From Theorem 5.5.9 and Theorem 5.6.7 it follows that the certainty equivalent in the
terminal wealth problem simplifies to
CEIT = x0
"
E[e(1−γ)f (0,IT ,0) ]
a=0
e(1−γ)f (0,0)
! γ1
#
−1 ,
where the function f is given in Theorem 5.5.9 and with
e
(1−γ)f a=0 (0,0)
=e
γrT
d Y
k=1
× exp
γ 0
d T
1+
γ −1 k
γ−1
−γ
1 + d0k T
! 12
(5.41)
!
−1 −1
1 γ
γ
⊤ −1
T (σλ0 − r1d ) σ
T Σ0
Id +
σ (σλ0 − r1d ) .
2 1−γ
γ −1
Now we determine the expectation appearing in CEIT
E[e(1−γ)f (0,IT ,0) ]
=e
γrT
d Y
k=1
"
× E exp
γ IT
d T
1+
γ −1 k
γ−1 1 + dIkT T
−γ
! 21
(5.42)
!#
1 γ
γ
⊤ −1
IT −1 −1
T (σ λ̂(0, IT , 0) − r1d ) σ
Id +
T Σa
σ (σ λ̂(0, IT , 0) − r1d ) .
2 1−γ
γ −1
In order to simplify the notation we use the abbreviation
d
X := σ −1 (σ λ̂(0, IT , 0) − r1d ) = N (b, B)
with b := σ −1 (σλ0 − r1d ), B := Σ0 − ΣIaT . Firstly, we consider only the expectation appearing in
(5.42), i.e.
"
!#
1 γ
γ
⊤ −1
IT −1 −1
E exp
T (σ λ̂(0, IT , 0) − r1d ) σ
Id +
T Σa
σ (σ λ̂(0, IT , 0) − r1d )
2 (1 − γ)
γ −1
Z
γ
1
1 γ
1
IT −1
⊤ −1
T x Id +
T Σa
x − (x − b) B (x − b) dx
exp
=
d p
2 (1 − γ)
γ −1
2
(2π) 2 det(B)
135
5 Continuous-Time Consumption-Investment Problems
q
−1
γ
γ
det B −1 + γ−1
T Id + γ−1
T ΣIaT )−1
p
=
det(B)
1
1
γ
γ
× exp − b⊤ B −1 b + b⊤ B −1 (B −1 +
T (Id +
T ΣIaT )−1 )−1 B −1 b
2
2
γ −1
γ −1
1
q
×
−1
d
γ
γ
(2π) 2 det B −1 + γ−1
T Id + γ−1
T ΣIaT )−1
Z
⊤
γ
γ
1
T (Id +
T ΣIaT )−1 )−1 B −1 b
× exp − x − (B −1 +
2
γ −1
γ −1
× B
= exp
−1
!
γ
γ
γ
γ
IT −1
−1
IT −1 −1 −1
T Id +
T Σa )
x−(B +
T (Id +
T Σa ) ) B b dx
+
γ −1
γ −1
γ −1
γ −1
−1
γ
γ
1
T Id +
T Σ0
b
− b⊤
2 γ −1
γ −1
!
d
Y
k=1
γ
1 + γ−1
T dIkT
γ
T d0k
1 + γ−1
! 21
.
γ
γ
T Id + γ−1
T ΣIaT )−1 is positive definite due to the assumptions of Theorem
Note that B −1 + γ−1
5.5.6. Hence, the last three lines of the second equality become one since the integrand together
with the square root can be interpreted as the density of a multivariate normally distributed
random variable. For the last equality we simply calculate the determinant using the eigenvalues
of ΣIaT and Σ0 . Consequently
E[e(1−γ)f (0,IT ,0) ]
!
−1 −1
1 −1
γ
⊤ γ
= exp γrT − (σ (σλ0 − r1d ))
T Id +
T Σ0
(σ (σλ0 − r1d ))
2
γ −1
γ −1
!γ − 12
γ
d
Y
dIkT T 2
1 + γ−1
γ 0
d T
.
×
1+
γ −1 k
1 + dI T T
k
k=1
Summarizing the above considerations we obtain


! 21
IT
γ
d
0
 Y (1 + γ−1 dk T )(1 + dk T )

CEIT = x0 
− 1 .
γ
0 T )(1 + dIT T )
d
(1
+
k
k=1
γ−1 k
Analogously, we get
CEXY =x0
"
Y
E[e(1−γ)f (0,IT ,0) ]
X ,0)
E[e(1−γ)f (0,IT
]
! γ1
#

− 1 = x0 
d
Y
k=1
γ
dYk T
1 + γ−1
1 + dYk T
1 + dX
k T
γ
dX
1 + γ−1
k T
! 21

− 1 .
In the same way as in Theorem 5.6.5, the last statement follows from part b) using that the
γ
function f (x) := (1 + γ−1
xT )(1 + xT )−1 is non-increasing for x > −T −1 .
Nevertheless, for the consumption-investment problem we are able to show that under some initial
conditions the investor is indifferent between the three types of insider information.
Theorem 5.6.9. Let Σ0 and Σε be simultaneously diagonalizable. Moreover, let a′ ∈ [0, 1) be the
Y
convex factor of ITX and let a ∈ [0, 1) be the convex factor of ITY . If ΣX
a′ = Σa , then the investor
X
Y
is indifferent between the insider information IT and IT (i.e. CEXY = 0).
136
5.6 Properties of the Continuous-Time Problems
Proof. It follows analogously to Theorem 4.6.9 (with the same transformation (4.34)) that under
the given assumptions
Z
Z
g X (0, iT , 0)q0X (iT )diT = g Y (0, iT , 0)q0Y (iT )diT ,
where q0X , q0Y are the Lebesgue densities of the distributions of ITX and ITY . Then by Theorem
5.6.7 we obtain the assertion.
5.6.2 Summarizing Remarks
For logarithmic utility functions the certainty equivalence principle holds, i.e. we obtain the
optimal investment strategy by replacing in the Merton proportion πM
πM := Σ−1 (σλ − r1d )
the unknown drift λ by the estimator λ̂(t, IT , Yt ) - this strategy is also known as plug-in strategy.
The optimal investment strategy depends linearly on the insider information IT and linearly on
the observation Yt . On the other hand, the optimal consumed fraction of the wealth is only a timedependent deterministic function which is without effect on the investment decisions. Hence, we
obtained in the terminal wealth problem (without consumption) and the consumption-investment
problem the same optimal investment strategy (see Theorem 5.5.2 and Theorem 5.5.4). The value
function depends quadratically on IT and Yt . Furthermore, we derived in Theorem 5.6.3 explicitly
the certainty equivalent between the three types of insider information. Then in Theorem 5.6.5
we found that for some parameterizations the investor prefers additional information about the
terminal stock prices rather than additional information about the drift or about the terminal
value of the Brownian motion.
For power utility functions the optimal consumption-investment strategy depends on IT and Yt .
γ
Here the certainty equivalence principle does not hold. Compared to the Merton proportion πM
(for power utility functions)
γ
:=
πM
1
Σ−1 (σλ − r1d )
1−γ
the unknown drift λ is again replaced by the estimator λ̂(t, IT , Yt ), however, an additional hedging
part is added. The latter depends on IT and Yt through an integral which is difficult to evaluate
(see Theorem 5.5.5). For the terminal wealth problem (without consumption) the additive hedging
part in the optimal investment strategy simplifies considerably (compare Theorem 5.5.9) and is
a linear function in IT and Yt . Whereas in the consumption-investment problem the certainty
equivalent is complicated to determine, in the terminal wealth problem we were able to present
the indifference price in closed form (see Corollary 5.6.8). Then for some parameterizations stock
price information has again a higher value than Brownian or drift information (see also Corollary
5.6.8).
137
6 Comparison of the Continuous-Time and
the Discrete-Time Problems
So far, the time lag h > 0 was fixed. Therefore, we used simplified notations as introduced in
Section 4.1. In this chapter we consider the value and the optimal strategy of the h-investor’s
consumption-investment problem for h tending to zero. We compare the corresponding limits
with the value and the strategy of the continuous-time problem. Hence, it is important to make
clear when a process depends on h. In order to distinguish the h-investor and the continuous-time
investor we mark all processes from Chapter 4 associated to the h-investor by a superscript h, for
instance
Znh
:= Zn = (Zn1 , . . . , Znd )⊤
Ynh
:= Yn =
n
X
Zkh
j
j
with Znhj := Znj = λj h + (Wnh
− W(n−1)h
), j = 1, . . . , d;
for the sum of observations;
k=1
Xnπh
:= Xnπ
for the wealth process under the admissible strategy π ∈ Ah0
for n = 0, . . . , N − 1 with T = N h, h ∈ [0, 1]. We denote the expected reward of the h-investor
under strategy π ∈ Ah0 by
"N
#
X
h
πh
V0π (x0 , iT ) := E
hUc (ck−1 ) + Up (XN ) | IT = iT .
k=1
h (x , i ) is the value of (P h ).
Then V0h∗ (x0 , iT ) := sup V0π
0 T
π∈Ah
0
From now on let Assumption 5.2.20 be satisfied.
The estimator of the insider drift
The h-investor estimates the drift and includes the insider information via the conditional dish
tribution (Q̂Z
n (· | iT , Yn ))n∈{0,...,N −1} . On the other hand, the continuous-time investor uses
(λ̂(t, iT , Yt ))t∈[0,T ] as the estimator for the drift containing the insider information. At time
points t = nh, n = 0, . . . , N, the continuous-time and the h-investor have the same observation
since
Yt = Ynh :=
n
X
Zkh
k=1
(Y is the observation process of the continuous-time investor) and moreover
Z
n
X
Zkh ).
hλ̂(t, IT , Yt ) = z Q̂Z
n (dz | IT ,
(6.1)
k=1
Hence, the h-insider and the continuous-time insider use the same estimator for the insider drift.
139
6 Comparison of the Continuous-Time and the Discrete-Time Problems
6.1 Continuous-Time Consumption-Investment Problems with
Constraints
In contrast to the continuous-time investor, for the h-insider it is neither optimal to short sell
a stock in order to buy a bond nor to borrow money from the bank in order to buy a stock.
Consequently, the optimal continuous-time strategies derived in the previous chapter are not
robust with respect to discretization. Therefore, we consider a continuous-time consumptioninvestment problem with short selling constraints
#
 "
RT


π


E Uc (ct )dt + Up (XT ) | IT = iT → max;

 0
(P̄ c ) Xtπ ≥ 0, t ∈ [0, T ];


π


X0 = x0 ;


π = (ct , αt )t∈[0,T ] ∈ A0 and αt ∈ D for all t ∈ [0, T ]
with D = {α ∈ Rd : 0 ≤ αk ≤ 1, k = 1, . . . , d, α⊤ 1d ≤ 1} and for A0 see Definition 5.1.3.
Remark 6.1.1. Due to the constraints on the portfolio strategy the associated HJB equation (see
(5.23) with controls u ∈ D instead of u ∈ Rd ) does even for logarithmic and power utility functions
not necessarily have a smooth solution (contrary to the unconstrained problems). Hence, we
might not be able to use standard verification arguments and one might have to look for viscosity
solutions. However, existence results for the optimal strategy can be derived by a dual approach
following Cvitanić & Karatzas [1992]. It is important to mention that we are not in exactly the
same setting as we have partial and insider information. The filtration of interest FIT Y is not
the Brownian filtration. Nevertheless, the approach of Cvitanić & Karatzas [1992] still works here
using Theorem 5.3.8 instead of standard martingale representation results (i.e. with Brownian
filtration).
We denote the value of (P̄ c ) by V̄0∗ (x0 , iT ). The set of admissible strategies for a continuous-time
investor without constraints is a superset of the admissible strategies of an investor with short
selling constraints. For all x0 > 0 and iT ∈ Rd it holds
V̄0∗ (x0 , iT ) ≤ V0∗ (x0 , iT ).
Moreover, for the continuous-time investor there exists an admissible strategy π̃ such that
V0h∗ (x0 , iT ) = V0π̃ (x0 , iT ). We illustrate this for the terminal wealth problem (without consumption). To this end, we suppose that the continuous-time investor acts according to the optimal
h∗ of his
strategy αh∗ . At time points t = nh the continuous-time investor invests the fraction αn
current wealth into the stock. Between the two trading time points tn = nh and tn+1 = (n + 1)h
the investor observes the financial market but keeps the number of stocks and bonds in his portfolio constant. However, due to the continuous changes in the stock price the invested fractions and
the current wealth change continuously. This leads to an adapted and càdlàg strategy α̃. The
latter is admissible for the continuous-time investor with short selling constraints (we illustrate
this strategy in Figure 6.1). Under this strategy the continuous-time investor attains the same
terminal wealth as the h-investor under αh∗ . Hence, the continuous-time investor is always able
to do better than the h-investor
V0h (x0 , iT ) ≤ V̄0∗ (x0 , iT ).
Figure 6.1 shows two possible paths of α̃ for the terminal wealth problem with logarithmic
utility function. The same plot contains the corresponding path of the stock price process. For
140
6.1 Continuous-Time Consumption-Investment Problems with Constraints
the simulation we choose the parameters as in the numerical example in Section 4.6 (page 62).
However, we set the time lag to h = 2−2 . Therefore, with T = 1 we have four trading time
points (N = 4). The blue dots correspond to the optimal investment strategy of the h-investor
h∗ , n = 0, . . . , 3. Between the trading time points the investment strategy (blue line) changes
αn
according to the changes in the stock price (red line). Even though the changes in the strategy
are clearly less significant, we observe similar movements in the strategy and in the stock.
2
2
h∗
αn
No trading
Stock price
1.5
1.5
1
1
0.5
0.5
0
0
0
0.5
t
1
0
0.5
1
t
Figure 6.1: Optimal discrete-time strategy when observing continuously (for fixed ω)
6.1.1 Dual Approach for Constrained Consumption-Investment Problems
In the unconstrained consumption-investment problem the insider market is FTIT Y -complete - in
the sense that every integrable and FTIT Y -measurable contingent claim can be replicated by an
admissible strategy π ∈ A0 . The classical dual approach is based on this market completeness
as it guarantees the existence of an optimal investment strategy (see e.g. Korn & Korn [2001]).
The classical market incompleteness occurs when the number of traded assets is smaller than the
number of market uncertainties. Even though the market is not incomplete in the classical sense
the presence of some convex constraints on the investment strategy can result in the inability of
replicating certain contingent claims. Hence, owing to the exclusion of short selling and borrowing
we cannot use the classical dual approach as in Korn & Korn [2001] together with the martingale
measure derived in Section 5.3 in order to solve (P̄ c ).
Remark 6.1.2. As shown by Cvitanić & Karatzas [1992] the classical incompleteness can be
viewed as being generated by imposing fictitious convex constraints: We can imaging to add some
fictitious assets which make the market complete in the classical sense, however, we then impose
some convex constraints which do not allow to trade in the fictitious assets.
According to Cvitanić & Karatzas [1992] the constrained consumption-investment problem (P̄ c )
can be embedded into a family of unconstrained problems (P ν ) in a "fictitious financial market",
where a so-called dual process ν ∈ H is used to describe the "incompleteness" of the market. Here
H is the set of all Rd -valued processes ν = (νt )t∈[0,T ] which are FIT Y -progressively measurable
141
6 Comparison of the Continuous-Time and the Discrete-Time Problems
and E
"
RT
0
#
νt⊤ νt + δ(νt )dt < ∞. The function δ is for x ∈ Rd the support function for the convex
set −D, i.e.
δ(x) := sup {−α⊤ x} = max{−x1 , . . . , −xd , 0}
α∈D
with its effective domain
D̃ := {x ∈ Rd : δ(x) < ∞} = Rd .
We summarize the main results of the dual approach for consumption-investment problems with
convex constraints introduced by Cvitanić & Karatzas [1992] and adapt them to our insider model
with filtration FIT Y . For the sake of notation simplicity we suppose here U (x) := Up (x) = Uc (x)
and denote by I the inverse of its derivative U ′ .
Auxiliary Unconstrained Consumption-Investment Problem in a Fictitious Financial Market
We set up the family of unconstrained auxiliary problems (P ν ) by introducing at first the fictitious financial market. For t ∈ [0, T ] and for any dual process ν ∈ H we define:
• The bond and the stock price processes
dBtν =Btν (r + δ(νt ))dt;



d
d
X
X
dStνk =Stνk 
σkj (λ̂(t, IT , Yt ))j + νtk + δ(νt ) dt +
σkj dWtIT Y j  ,
j=1
k = 1, . . . , d,
j=1
where (λ̂(t, IT , Yt ))j denotes the j-th component of the Rd -valued insider drift.
• The discounted exponential local martingale
 t

Z
Zt
1 ν ⊤ ν
Htν := exp − (θsν )⊤ dWsIT Y −
(θ ) θs + r + δ(νs )ds ,
2 s
0
0
where θtν := σ −1 (σ λ̂(t, IT , Yt ) − r1d ) + σ −1 νt is the market price of risk.
With the fictitious price processes of the assets the wealth process under a strategy π ∈ A0 is
given by
dXtνπ = Xtνπ (r + δ(νt ) + αt⊤ (νt + σ λ̂(t, IT , Yt ) − r1d ) − ct dt + Xtνπ αt⊤ σdWtIT Y , X0νπ = x0 ,
and by Itô’s formula
Htν Xtνπ +
Zt
0
Hsν cs ds = x0 +
Zt
0
Hsν Xsνπ (αs⊤ σ − (θsν )⊤ )dWsIT Y .
(6.2)
The family of auxiliary unconstrained consumption-investment problems can then be written as
#
 "
RT



E U (ct )dt + U (XTνπ ) | IT = iT → max;



 0
(P ν ) Xtνπ ≥ 0, t ∈ [0, T ];



X0νπ = x0 ;



π = (ct , αt )t∈[0,T ] ∈ A0 .
142
6.2 Convergence Results
The existence of an optimal consumption-investment strategy of (P ν ) follows from Cvitanić &
Karatzas [1992]. For this purpose, we define at first
 T

Z
F (y, iT ) := E  Htν I(yHtν )dt + HTν I(yHTν ) | IT = iT  .
0
Let us suppose F to be finite for all y > 0 and iT ∈ I. Then for all fixed iT ∈ I the function
F (y) := F (y, iT ) is continuous and strictly decreasing over (0, ∞). Moreover, lim F (y) = ∞ and
yց0
lim F (y) = 0 (compare Korn & Korn [2001] Lemma 3 Chapter V). Hence, there exists a unique
y→∞
y ∗ > 0 with
F (y ∗ ) = x0 .
In the following lemma we briefly formulate the existence results for an optimal solution of (P ν ) in
part a) and part b). In part c) we then establish the connection between the optimal consumptioninvestment strategies of (P ν ) and (P̄ c ) (compare Cvitanić & Karatzas [1992] Proposition 8.3).
Lemma 6.1.3.
a) Let cνt := I(y ∗ Htν ) and C ν := I(y ∗ HTν ). Then cνt and C ν satisfy
 T

 T

Z
Z
E  U (cs )ds + U (XTνπ )|F0IT Y  ≤ E  U (cνs )ds + U (C ν )|F0IT Y 
0
0
for all π = (ct , αt )t∈[0,T ] ∈ A0 for which
Xtνπ
≥ 0.
b) Let cν = (cνt )t∈[0,T ] be a consumption strategy and C ≥ 0 an FTIT Y -measurable random
variable with


ZT
x0 = E HTν C + Hsν cνs ds|F0IT Y 
0
for any dual process ν ∈ H. Then there exists an investment strategy αν = (αtν )t∈[0,T ] such
ν
ν
that π ν = (cνt , αtν )t∈[0,T ] ∈ A0 with Xtνπ ≥ 0 and XTνπ = C.
ν∗
c) For some ν ∈ H let π ν∗ = (cν∗
t , αt )t∈[0,T ] be an optimal consumption-investment strategy
ν
of (P ). Furthermore, let for all t ∈ [0, T ]
αtν∗ ∈ D
and
δ(νt ) + (αtν∗ )⊤ νt = 0.
Then π ν∗ is an optimal consumption-investment strategy for the constrained problem (P̄ c ).
Remark 6.1.4. Lemma 6.1.3 follows (directly) from Cvitanić & Karatzas [1992] Theorem 7.4
together with Lemma 7.2 and Proposition 7.3. However, in order to prove part b) we have to use
Theorem 5.3.8 instead of standard martingale representation results (i.e. with Brownian filtration)
as used in Cvitanić & Karatzas [1992] (see Lemma A.2.1).
6.2 Convergence Results
In this section we consider an investor with logarithmic or power utility functions. For both utility
functions, we state at first existence results for the constrained problem (P̄ c ) and investigate then
the behavior of (P h ) for the time lag h tending to zero.
143
6 Comparison of the Continuous-Time and the Discrete-Time Problems
6.2.1 Logarithmic Utility Functions
We consider an investor with logarithmic utility functions, i.e. we suppose U (x) = log(x).
Constrained Consumption-Investment Problem
Theorem 6.2.1. The optimal consumption-investment strategy of problem (P̄ c ) is given by
X
π
t
and αt is the unique solution of the (pointwise) quadratic
π = (ct , αt )t∈[0,T ] where ct := T −t+1
maximization problem (Q)
(
α⊤ (σ λ̂(t, IT , Yt ) − r1d ) − 12 α⊤ Σα → max
(Q)
α ∈ D.
Remark 6.2.2. (Q) is a standard quadratic optimization problem with a unique optimal solution
(since D , ∅ and Σ positive definite). For d = 1 we just have to solve the corresponding unconstrained maximization problem (i.e. α ∈ R) and shift the optimal solution to the nearest boundary
of D = [0, 1] if the unconstrained solution is not in D, i.e.


if Σ−1 (σ λ̂(t, IT , Yt ) − r1d ) < 0;
0,
−1
αt = Σ (σ λ̂(t, IT , Yt ) − r1d ), if Σ−1 (σ λ̂(t, IT , Yt ) − r1d ) ∈ [0, 1];


1,
else.
For d > 1 this is in general not true; but the optimal solution can at least be characterized by
the Kuhn-Tucker conditions. By a measurable selection theorem it follows that the pointwise
maximization in (Q) leads indeed to an admissible strategy which is FIT Y -progressively measurable
(see Cvitanić & Karatzas [1992]).
Proof. For U (x) = log(x) we have U ′ (x) = x1 , hence I(y) = y1 . The conditions of Lemma 6.1.3
νπ
part a) and b) imply cs = (y ∗ Htν )−1 and XT = (y ∗ HTν )−1 with y ∗ being the solution of F (y ∗ ) =
RT
ν X νπ + H ν c ds = x . Thus, the non-negative local martingale
x0 . Hence, y ∗ = Tx+1
and
H
0
s s
T
T
0
0
νπ
(Xt Htν
+
Rt
0
Hsν cs ds)t∈[0,T ]
x0 =
X
νπ
is even a true martingale, i.e.

νπ
E XT HTν
+
ZT
Hsν cs ds|FtIT Y
0

 = Xtνπ Htν +
νπ
t
. By Lemma 6.1.3 part c) Xt
and thus ct = T −t+1
fractions of the wealth are given by
c̃t =
Zt
Hsν cs ds
π
= Xt such that the optimal consumed
1
.
T −t+1
Furthermore, a comparison of (6.3) with (6.2) yields
αt =Σ−1 (νt + (σ λ̂(t, IT , Yt ) − r1d )),
where νt is the solution to the following pointwise minimization problem
νt := arg min {2δ(x) + ||σ −1 (σ λ̂(t, IT , Yt ) − r1d ) + σ −1 x||2 }
x∈Rd
144
(6.3)
0
6.2 Convergence Results
(see Cvitanić & Karatzas [1992] Section 11 and Section 10 (D)). Here || · || denotes the Euclidean
norm. Using duality results for convex programming we show that αt solves the quadratic maximization problem (Q). The support function δ(x) = sup {−z ⊤ x} represents itself a linear opz∈D
timization problem with linear inequality constraints. We denote this optimization problem by
(P δ ) and the corresponding dual one by (Dδ )


⊤
⊤


z x → min
y b → min
(P δ ) Az ≤ b
(Dδ ) x = −A⊤ y




d
y ∈ Rd+1
z∈R
≥0
with A and b defined by




A := 


−1
1
0
−1
1
...
..
.
...
0




 ∈ R(d+1)×d ,

−1
1
 
0
 .. 
 
b :=  .  ∈ Rd+1 .
0
1
As the sets of admissible points of the linear optimization problems (Dδ ) and (P δ ) are non-empty,
the strong duality holds. Hence, (Dδ ) and (P δ ) have optimal solutions and the optimal values
δ
⊤
⊤
are identical. Let y ∈ Rd+1
≥0 be the optimal solution of (D ) with value y b and x = −A y. Then
νt = −A⊤ arg min {2y ⊤ b+ || σ −1 (σ λ̂(t, IT , Yt ) − r1d ) − σ −1 A⊤ y ||2 }.
y∈Rd+1
≥0
On the other hand, the dual problem (D) corresponding to (Q) is given by

1 ⊤
⊤

− 2 α Σα − y b → max
(D) α = Σ−1 (−A⊤ y + σ λ̂(t, IT , Yr ) − r1d )


α ∈ Rd , y ∈ Rd+1
≥0 .
Since Σ is positive definite there exists an optimal solution. If (α∗ , y ∗ ) is an optimal solution of
(D), then α∗ is an optimal solution of (Q) by strong duality (for convex programming). Moreover,
the optimal values of (Q) and (D) are identical. Hence
α∗ = Σ−1 (−A⊤ arg min {2y ⊤ b + ||σ −1 (σ λ̂(t, IT , Yt )−r1d )−σ −1 A⊤ y||2 }+σ λ̂(t, IT , Yt )−r1d ),
y∈Rd+1
≥0
i.e. α∗ = αt .
Let π = (ct , αt )t∈[0,T ] be the optimal strategy of (P̄ c ) as given in Theorem 6.2.1. Then
∗
V 0 (x0 , iT ) = V0π (x0 , iT ).
Let c̃t be the optimal consumed fraction of the wealth, i.e. c̃t =
time strategy π h = (chn , αhn )n∈{0,...,N −1} defined by
πh h
• chn := c̃hn Xn
• αhn := αnh .
with c̃hn := c̃nh =
1
T −t+1 .
We consider the discrete-
1
;
(N − n)h + 1
(6.4)
145
6 Comparison of the Continuous-Time and the Discrete-Time Problems
The strategy π h depends only through λ̂(t, IT , Yt ), t = nh, on the past. Hence, π h is - thanks to
(6.1) - an admissible strategy for the h-insider. Note that αh corresponds at trading time points
t = nh to the optimal investment decision of the continuous-time investor with short selling
constraints. Due to the a.s.-continuity of t 7→ π t the simple process
αh0 1{0} (t) +
⌊T
h⌋
X
αhk−1 1((k−1)h,kh] (t)
k=1
convergences almost surely to αt as h → 0.
Convergence of the Value
Theorem 6.2.3. With the consumption-investment strategies π and π h as defined in (6.4) it
holds that
lim V h h (x0 , iT ) = V̄0π (x0 , iT )
h→0 0π
= V̄0∗ (x0 , iT )
for all x0 > 0 and iT ∈ Rd .
Proof. The second equality holds by definition. In order to prove the first equality we consider
the wealth recursion under strategy π h and the wealth SDE under strategy π. We start with the
wealth recursion under π h , i.e.
h
πh h
πh h
1 + (αhn−1 )⊤ (e−rh R(Znh ) − 1)
Xn =erh Xn−1 1 −
(N − (n − 1))h + 1
n
Y
(N − k)h + 1 ⊤
1 + αhk−1 (e−rh R(Znh ) − 1)
=x0 erhN
(N − (k − 1))h + 1
k=1
with corresponding expected reward
h
V0π
h (x0 , iT )
(6.5)
=N h log(x0 ) −
+
N
X
k=1

N
X
k=1

E h log 
+ log(x0 ) +
N
X
k=1
+E
"
N
X
h log((N − (k − 1))h + 1)
k−1
Y
j=1
(N − k)h + 1
log
(N − (k − 1))h + 1
log 1 +
k=1


(N − j)h + 1
erh 1 + (αhj−1 )⊤ (e−rh R(Zjh ) − 1d )  |IT = iT  + rN h
(N − (j − 1))h + 1
(αhk−1 )⊤ (e−rh R(Zkh ) − 1d )
|IT = iT
rT
1
h − (T + 1) log(T + 1)
=(T + 1) log(x0 ) + rT ( T + 1) −
2
2
N
h i
X
E log 1 + (αhk−1 )⊤ (e−rh R(Zkh ) − 1d ) |IT = iT
+
#
k=1
+
N
k−1
i
X
X h h
E log 1 + (αhj−1 )⊤ (e−rh R(Zjh ) − 1d ) |IT = iT .
k=1
146
j=1
6.2 Convergence Results
The wealth SDE under π is given by
π
π
dXt = Xt r + α⊤
t (σ λ̂(t, IT , Yt ) − r1d ) −
Applying Itô’s formula shows that
 t
Z π

r + α⊤
Xt = x0 exp
s (σ λ̂(s, IT , Ys ) − r1d ) −
0
Hence, we can write the value of
(P̄ c )
as
1
π
IT Y
.
dt + Xt α⊤
t σdWt
T −t+1

Zt
1
1 ⊤
IT Y 
.
− α Σαs ds + α⊤
s σdWs
T −s+1 2 s
0
V̄0∗ (x0 , iT )
(6.6)

 T
Z
1
1
⊤

=rT ( T + 1) + (T + 1) log(x0 ) + E  α⊤
t (σ λ̂(t, IT , Yt ) − r1d ) − αt Σαt dt|IT = iT
2
2
0
 T s

Z Z
1 ⊤

α⊤
+E
u (σ λ̂(u, IT , Yu ) − r1d ) − αu Σαu duds + |IT = iT − (T + 1) log(T + 1).
2
0 0
Comparing (6.5) and (6.6) it remains to show that


N
X
i
h 
lim 
E log 1 + (αhk−1 )⊤ (e−rh R(Zkh ) − 1d ) |IT = iT
h→0 
|k=1
{z
}
(I)


k−1
N
i
X h X

h
⊤ −rh
h
E log 1 + (αj−1 ) (e
h
+
R(Zj ) − 1d ) |IT = iT 

k=1 j=1

|
{z
}
(II)

 T
Z
1 ⊤

=E  α⊤
t (σ λ̂(t, IT , Yt ) − r1d ) − π t Σαt dt|IT = iT
2
0

 T s
Z Z
1
⊤

α⊤
+E
u (σ λ̂(u, IT , Yu ) − r1d ) − αu Σαu duds|IT = iT .
2
0 0
Now we adopt the ideas of Theorem 5.1 in Bäuerle et al. [2012]. At a fixed time point t =
IT Z
(n − 1)h ∈ [0, T ] and given the information Fn−1
we know αhn−1 . Therefore, we treat αhn−1 =
hd ⊤
(αh1
n−1 , . . . , αn−1 ) as a constant and expand




exp(Z̃n1 )




..
f (Z̃n1 , . . . , Z̃nd ) := log 1 + (αhn−1 )⊤ 
 − 1d 
.
exp(Z̃nd )
with Z̃nk :=
d
P
j=1
σkj Znhj − (r + 21
d
P
j=1
2 )h, k = 1, . . . , d, in a Taylor series around 0
σkj
d×1 . It follows
from the Taylor expansion and from the tower property of the conditional expectation (see Lemma
147
6 Comparison of the Continuous-Time and the Discrete-Time Problems
A.2.3) that
i
h E log 1 + (αhn−1 )⊤ (e−rh R(Znh ) − 1d ) |IT
3
1
h
⊤
h
⊤
h
=E h(αn−1 ) (σ λ̂((n − 1)h, IT , Y(n−1)h ) − r1d ) − h(αn−1 ) Σαn−1 |IT + o(h 2 ).
2
3
3
In the remainder o(h 2 ) we collect all terms which can be bounded by h 2 times an integrable
and σ(IT )-measurable random variable - let us say ζ. The random variable ζ can be chosen
3
3
independent of h and n such that |o(h 2 )| ≤ h 2 ζ a.s. and
we obtain for the first term (I)
⌊T
h⌋
P
k=1
3
3
1
o(h 2 ) = ⌊ Th ⌋o(h 2 ) = o(h 2 ). Thus,
⌊T
h⌋
i
X h E log 1 + (αhk−1 )⊤ (e−rh R(Zkh ) − 1d ) |IT
(6.7)
k=1

⌊T
h ⌋
=E 
X
k=1

1
1
(αhk−1 )⊤ (σ λ̂((k − 1)h, IT , Y(k−1)h )−r1d )h − (αhk−1 )⊤ Σαhk−1 h +o(h 2 )|IT  .
2
Analogously, we obtain for the second term (II)
⌊T
h⌋
i
X k−1
X h h
E log 1 + (αhj−1 )⊤ (e−rh R(Zjh ) − 1d ) |IT
k=1

(6.8)
j=1

X
X k−1
1
1
(αhj−1 )⊤ (σ λ̂((j − 1)h, IT , Y(j−1)h ) − r1d )h− (αhj−1 )⊤ Σαhj−1 h +o(h 2 )|IT  .
h
=E 
2
⌊T
h⌋
k=1
j=1
Due to the boundedness of αh and the a.s.-continuity of t 7→ λ̂(t, IT , Yt ) on [0, T ] we can find a
random variable ξ (independent of h and k) such that
T
⌊T
⌊ h ⌋
h⌋
X
X h ⊤
1
h
h
⊤
(αk−1 ) (σ λ̂((k − 1)h, IT , Y(k−1)h ) − r1d )h −
(αk−1 ) Σαk−1 h ≤ ξ a.s.
2
k=1
k=1
For instance, we choose ξ as ξ := T λ̂∗T + c for a constant c ∈ R≥0 and λ̂∗T := max |1⊤
d σ λ̂(t, IT , Yt )|.
t∈[0,T ]
(Id + tΣIaT )−1 (CλIT λ0 + C IT IT
+ ΣIaT (λt + Wt ))
the integrability of ξ follows
Since λ̂(t, IT , Yt ) =
e.g. by Doob’s maximal inequality. Hence, we can apply dominated convergence in order to
interchange the order of the limit and the expectation in (I) and (II). Taking the limits of the
arguments of the expectations in (6.7) and in (6.8) as done in Lemma A.2.5 completes the proof.
By virtue of the inequality
h
h∗
∗
V0π
h (x0 , iT ) ≤ V0 (x0 , iT ) ≤ V̄0 (x0 , iT )
we obtain directly from Theorem 6.2.3 the following result which states that the discretized
optimal continuous-time strategy (with short selling constraints) π h is a good and simple strategy
for the h-investor.
148
6.2 Convergence Results
Corollary 6.2.4. With the consumption-investment strategies π and π h as defined in (6.4) it
holds:
a) lim V0h∗ (x0 , iT ) = V̄0∗ (x0 , iT );
h→0
h (x , i )| = 0
b) lim |V0h∗ (x0 , iT ) − V0π
0 T
h
h→0
for all x0 > 0 and iT ∈ I.
Convergence of the Optimal Strategy
At discrete time points t = nh the optimal consumed fractions of the continuous-time insider
and the h-insider are identical and given explicitly. On the other hand, the optimal investment
strategy of the h-insider cannot be given in closed form and has to be computed numerically.
However, in Theorem 6.2.5 we show that the optimal investment strategy of the h-investor is
close to the discretization of α (given in Theorem 6.2.1). To this end, we define φh = (φht )t∈[0,T ]
as a piecewise constant investment strategy which corresponds at discrete time points t = nh, n =
0, . . . , N − 1, to the optimal investment strategy αh∗ of the h-investor, i.e. φh is the piecewise
constant interpolation of the optimal discrete-time investment strategy, namely


⌊ ht ⌋
X
Zkh  .
(6.9)
φht := α⌊h∗t ⌋ IT ,
h
k=1
The process φh is càdlàg, FIT Y -adapted and admissible for the continuous-time investor (with
short selling constraints).
Theorem 6.2.5. For the strategy φh defined in (6.9) it holds:
lim φht = αt
h→0
a.s.,
t ∈ [0, T ].
Proof. According to Theorem 6.2.1 αt is the unique maximizer of
1
α 7−→ α⊤ (σ λ̂(t, IT , Yt ) − r1d ) − α⊤ Σα
2
h∗ is for t = nh the unique maximizer of
over α ∈ D. Due to Theorem 4.5.1 αn
Z
α 7−→ log 1 + α⊤ (e−rh R(z) − 1d ) Q̂Z
n (dz | IT , Yt )
over α ∈ D. The assertion follows then from Lemma A.2.6 together with Lemma A.3.3.
Figure 4.6, Figure 6.2 and Figure 6.3
In Figure 4.6 in Section 4.6 we plot the average optimal investment strategy of the h-investor
against the time lag h for fixed n = 9 and various values of the observation y, i.e.
Z
α9h∗ (iT , y)Q̂I9T (diT | y).
The same plot (dotted lines) contains the average approximated strategy (see (6.4)), i.e.
Z
αh9 (iT , y)Q̂I9T (diT | y).
149
6 Comparison of the Continuous-Time and the Discrete-Time Problems
Information iT = −0.25
Information iT = 0
1
1
y=1
0.8
y = 0.25
0.6
0.6
y=0
∗
(iT , y)
αn
∗
αn
(iT , y)
0.8
y = 0.5
y = −0.5
0.4
0.2
0
0.4
0.2
0
0
0.5
0
1
0.5
1
h
Information iT = 0.5
1
1
0.8
0.8
0.6
0.6
∗
αn
(iT , y)
∗
αn
(iT , y)
h
Information iT = 0.25
0.4
0.2
0
0.4
0.2
0
0
0.5
1
0
0.5
h
1
h
Figure 6.2: Optimal strategy in dependence on the time lag h for different realized values of y
and iT
Moreover, in Figure 6.2 we plot α9h∗ (iT , y) and αh9 (iT , y) for various realized values of the stock
price information (parameters are chosen as in Figure 4.6). In both figures we observe that even
for larger values of the time lag h the (average) optimal and the (average) approximated strategies
are almost identical. In the right column of Figure 6.3 we simulate for an investor with stock
price information two paths of the investment strategy α (green line) and the corresponding
paths of φh for time lags h ∈ {2−1 , 2−5 , 2−7 } (remaining parameters chosen according to the
numerical example in Section 4.6 on page 62). When h becomes smaller we observe that the
piecewise constant strategy φh tends to the optimal strategy α (for fixed t). The left column of
this figure contains the corresponding path of the stock price process (red line) as well as the
path of strategy α for an investor without insider information (magenta line). Even when less
significant, the strategy changes in the same direction as the stock price. It is intuitively plausible
to have a higher stock position when the stock prices are higher. However, for an investor with
stock price information we observe a reversed behavior. This unexpected behavior results in this
numerical example from the negative value of ΣIaT .
Remark 6.2.6. Let the consumption-investment strategy ϕ = (ϕt )t∈[0,T ] be defined by ϕt :=
X
ϕ
( T −⌊ t t⌋h+1 , φht )t∈[0,T ] . Here Xtϕ is the wealth under strategy ϕ which satisfies
h
dXtϕ
150
=
Xtϕ
r + (φht )⊤ (σ λ̂(t, IT , Yt ) − r1d ) −
1
t
T − ⌊ h ⌋h + 1
!
dt + Xtϕ (φht )⊤ σdWtIT Y .
6.2 Convergence Results
1
2.5
0.8
2
0.6
1.5
−1
φh
t ,h = 2
0.4
−5
φh
t ,h = 2
1
−7
φh
t ,h = 2
αt
0.2
0
0.5
0
0.5
0
1
t
0.5
1
t
1.4
0.8
Stock price
1.2
0.6
αt (a = 0)
1
0.4
0.8
0.2
0.6
0
0.4
0
0.5
1
0
t
0.5
1
t
Figure 6.3: Convergence of the optimal strategy for a fixed ω ∈ Ω
With Theorem 6.2.5 and dominated convergence we obtain
lim V0φh (x0 , iT ) = V̄0∗ (x0 , iT )
h→0
(6.10)
for all x0 ∈ R>0 and iT ∈ I.
6.2.2 Power Utility Functions
We consider an investor with power utility functions, i.e. U (x) = xγ , 0 < γ < 1. The existence
of an optimal consumption-investment strategy in case of power utility functions follows from
Theorem 13.1 in Cvitanić & Karatzas [1992] together with Remark 6.1.1. However, the solution
is not known in closed form. As remarked by Cvitanić & Karatzas [1992] it is doubtful that
explicit solutions exist for utility functions other than the logarithmic function.
An Admissible Strategy
We consider a continuous-time strategy π = (c̃t , αt )t∈[0,T ] , where c̃ is the fraction of the wealth
which is consumed. We suppose π to be FIT Y -adapted with t 7→ π t = (c̃t , αt ) càglàd or càdlàg,
151
6 Comparison of the Continuous-Time and the Discrete-Time Problems
0 ≤ c̃ ≤ 1 and α ∈ D. Then π is admissible for the continuous-time investor with short selling
constraints. We define the simple process π h = (c̃ht , αht )t∈[0,T ] by
•
c̃ht := c̃0 1{0} (t) +
•
αht
⌊T
h⌋
X
c̃(k−1)h 1((k−1)h,kh] (t);
(6.11)
k=1
:= α0 1{0} (t) +
⌊T
h⌋
X
α(k−1)h 1((k−1)h,kh] (t).
k=1
πh
Firstly, we suppose that π is càglàd. Then
is bounded and predictable. Moreover, by virtue of
the left continuity, π ht = (c̃ht , αht ) converges pointwise (in ω) to π t = (c̃t , αt ) as h → 0. When π is
càdlàg then π − is càglàd and π ht converges pointwise to (π − )t as h → 0 (π − denotes the process
whose value at time t is given by (π − )t := lim π s ). Furthermore, we denote the "wealth"
s→t,s<t
recursion under the discrete strategy (c̃nh , αnh )n∈{0,...,N −1} by X h = (Xnh )n∈{0,...,N −1} , i.e.
Xnh = x0 erhn
n
Y
k=1
−rh
,
e
R(Z
)
−
1
(1 − c̃(k−1)h h) 1 + α⊤
k
d
(k−1)h
X0h = x0 .
IT Y
The random variable (c̃nh , αnh ) is Fnh
-measurable but not necessarily FnIT Z -measurable (FnIT Z ⊆
IT Y
Fnh ). Hence, the strategy (c̃(n−1)h , α(n−1)h )n∈{0,...,N −1} needs not to be an admissible strategy for the h-investor. In that case X h is not a wealth process of the h-investor. As before
π
X π = (Xt )t∈[0,T ] is the continuous-time wealth process under strategy π, i.e.

 t
Zt
Z
1
π
⊤
IT Y 
.
α⊤
Xt =x0 exp  α⊤
s (σ λ̂(s, IT , Ys ) − r1d ) + r − c̃s − αs Σαs ds +
s σdWs
2
0
0
The piecewise constant strategy π h is also an admissible strategy for the continuous-time investor.
Even though the strategy does not change over a time interval of length h the investor has to trade
πh
h
continuously in order to keep the fractions constant. The process X π = (Xt )t∈[0,T ] denotes
the corresponding wealth process, i.e.

 t
Zt
Z
1
πh
Xt =x0 exp  (αhs )⊤ (σ λ̂(s, IT , Ys ) − r1d ) + r − c̃hs − (αhs )⊤ Σαhs ds + (αhs )⊤ σdWsIT Y  .
2
0
0
Convergence of the Value
π
πh
We look at the relation between Xnh , Xt and Xt .
Lemma 6.2.7. Let π = (π t )t∈[0,T ] be given as in (6.11). Then for t ∈ [0, T ] it holds:
⌊ ht ⌋
X
k=1
−rh
R(Zkh ) − 1d )
log (1 − c̃(k−1)h h) 1 + α⊤
(k−1)h (e
ucp
−→
Zt
0
α⊤
s (σ λ̂(s, IT , Ys ) − r1d ) −
1 ⊤
α Σαs − c̃s ds +
2 s
(6.12)
Zt
IT Y
α⊤
s σdWs
0
as h → 0 (h < 1). We denote by ucp the uniform convergence on compacts in probability.
152
6.2 Convergence Results
Remark 6.2.8. Since convergence in probability is preserved under continuous maps it follows
from Lemma 6.2.7 that
π
ucp
a) X⌊ht ⌋ −→ Xt ;
h
π h ucp
π
−→ Xt
b) Xt
for t ∈ [0, T ] as h → 0.
Proof. Let us first suppose that π is càglàd. Then the simple process π h in (6.11) is predictable and converges pointwise to π t . For n = 1, . . . , N the investment and consumption decision
IT Y
IT Y
-measurable (opposed to R(Zkh ) which is Fnh
-measurable). There(c̃(n−1)h , α(n−1)h ) is F(n−1)h
fore, we consider π (n−1)h = (c̃(n−1)h , α(n−1)h ) as constant and obtain with the Taylor expansion
of
⊤
f (x1 , . . . , xd ) := log 1 + α⊤ exp(x1 ) . . . exp(xd ) − 1d
around x = 0 the following representation (compare Theorem 6.2.3 and Lemma A.2.2)
−rh
R(Znh ) − 1d )
log (1 − c̃(n−1)h h) 1 + α⊤
(n−1)h (e
=
d
X
i=1
where Z̃nk :=
by |r3n | ≤
d
P
j=1
d
P
i,j,k=1
αi(n−1)h Z̃ni −
d
d
1 X i
1X i
α(n−1)h αj(n−1)h Z̃ni Z̃nj +
α(n−1)h (Z̃ni )2 + r3n ,
2
2
i,j=1
σkj Znhj − (r + 21
d
P
j=1
i=1
2 )h, k = 1, . . . , d. The remainder of third degree is bounded
σkj
|Z̃ni Z̃nj Z̃nk |. We also expand log(1 − c̃(n−1)h h) with c̃(n−1)h ∈ [0, 1], h < 1, in a
Taylor series around zero and obtain
log(1 − c̃(n−1)h h) = −hc̃(n−1)h + r2n
with |r2n | ≤
⌊ ht ⌋
X
n=1
h
1−h
2
(independent of n). Hence,
−rh
log (1 − c̃(n−1)h h) 1 + α⊤
R(Znh ) − 1d )
(n−1)h (e
(6.13)


⌊ ht ⌋ 
d
X X

d
d

1 X i
1X i

j
n 
i
i
i j
i 2
=
α(n−1)h α(n−1)h Z̃n Z̃n +
α(n−1)h (Z̃n ) − hc̃(n−1)h + r3 
α(n−1)h Z̃n −


2
2
| {z } |{z}
n=1  i=1
i,j=1
i=1

(V)
|
{z
}
|
{z
}
|
{z
}
(IV)
(I)
+o
h
(1 − h)2
(II)
(III)
.
We rewrite Znh using the observation process Y of the continuous-time investor
Znh = Ynh − Y(n−1)h .
153
6 Comparison of the Continuous-Time and the Discrete-Time Problems
Due to Theorem 5.2.22 the process Y is a continuous d-dimensional FIT Y -semimartingale with
unique decomposition
Yt = At + Mt ,
with
At :=
Zt
Mt := WtIT Y .
λ̂(s, IT , Ys )ds,
0
In order to ease the notation we simply write here Wtj := WtIT Y j , j = 1, . . . , d. We consider (I)-(V)
separately:
(I) We start with the first sum, namely
⌊ ht ⌋ d
XX
αi(n−1)h Z̃ni
n=1 i=1
=
⌊t⌋
d X
h
X
αi(n−1)h
i=1 n=1
d
X
j=1
j
j
σij (Ynh
−Y(n−1)h
)−
⌊t⌋
d X
h
X
d
αi(n−1)h (r+
i=1 n=1
1X 2
σij )h.
2
j=1
A càglàd process has only countably many jumps. Hence, the bounded process α has finitely
many discontinuities on [0, T ] and is therefore Riemann integrable. It follows that the last
sum in (I) (Riemann sums) converges pointwise to
Zt
0
d
αit (r +
1X 2
σij )dt.
2
j=1
We consider the first sum in (I). Using the simple process π h in (6.11) we can equivalently
write
⌊ ht ⌋
X
n=1
j
j
αi(n−1)h σij (Ynh
− Y(n−1)h
)=
Zt
j
αhi
t σij dYt
0
for i, j ∈ {1, . . . , d}. Then

2 
s
Z

hi
i
j 
E  sup (αu − αu )σij dYu 
s∈[0,t] 0

s
2 
Z
Zs

i
j
hi
i
j 
=E  sup (αhi
(α
−
α
)σ
dA
+
−
α
)σ
dW
u
u 
u
u ij
u
u ij
s∈[0,t] 0
0



 t
2
2 
 s
Z
Z



i
j 
hi
i
j
≤2E  sup  (αhi
 + 2E  |(αu − αu )σij ||dAu |  .
u − αu )σij dWu
s∈[0,t]
0
0
The term with the integral w.r.t. the finite variation process (on [0, T ]) tends to zero as
h → 0 by dominated convergence. Furthermore, with Doob’s maximal inequality and the
Itô isometry we obtain

s
2 
 t

Z
Z

i
j 
i 2 2
 (αhi

E  sup (αhi
u − αu )σij dWu  ≤ 4E
u − αu ) σij du .
s∈[0,t] 0
154
0
6.2 Convergence Results
Due to the boundedness of α the last expression tends to zero by dominated convergence.
Note that L2 -convergence implies convergence in probability. Hence, (I) tends in ucp to
Zt
IT Y
α⊤
s σdWs
+
Zt
0
0
α⊤
s σ λ̂(s, IT , Ys )ds −
Zt X
d
d
αis (r +
0 i=1
1X 2
σij )ds.
2
j=1
as h → 0.
(II) We split the second sum, i.e.
⌊ ht ⌋
d
X X
αi(n−1)h αj(n−1)h Z̃ni Z̃nj
n=1 i,j=1
=
⌊ ht ⌋
d
X
X
αi(n−1)h αj(n−1)h (r +
n=1
i,j=1
d
d
⌊t⌋
k=1
k=1
n=1
d
d
⌊t⌋
k=1
k=1
n=1
d
d
k=1
k=1
1X 2
1X 2 2
σjk )(r +
σik )h
2
2
h
X
1X 2 X
k
k
− Y(n−1)h
)
− (r +
σik )h
αi(n−1)h αj(n−1)h (Ynh
σjk
2
h
X
X
1X 2
k
k
− (r +
σik
σjk )h
− Y(n−1)h
)
αi(n−1)h αj(n−1)h (Ynh
2
+
d
X
⌊ ht ⌋
σjl σik
k,l=1
k,l
+
d
X
k=1
X
n=1
⌊ ht ⌋
σjk σik
X
n=1
k
k
l
l
− Y(n−1)h
)(Ynh
− Y(n−1)h
)
αi(n−1)h αj(n−1)h (Ynh
k
k
− Y(n−1)h
)2
αi(n−1)h αj(n−1)h (Ynh
!
.
(1)
(2)
(3)
(4)
(5)
The first term (1) tends obviously pointwise to zero as h → 0 (Riemann integrable). Since
Zt
0
hj
i j
k
(αhi
s αs − αs αs )dYs
converges in ucp to zero (see (I)), (2) and (3) will tend to zero in ucp. We rewrite (4) (and
(5)):
⌊ ht ⌋
X
k
k
l
l
− Y(n−1)h
αi(n−1)h αj(n−1)h (Ynh
)(Ynh
− Y(n−1)h
)
X
l
k
l
k
αi(n−1)h αj(n−1)h (Ynh
Ynh
− Y(n−1)h
Y(n−1)h
)
n=1
=
⌊ ht ⌋
n=1
−
⌊ ht ⌋
X
n=1
k
l
l
αi(n−1)h αj(n−1)h Y(n−1)h
(Ynh
−Y(n−1)h
)−
⌊ ht ⌋
X
n=1
l
k
k
αi(n−1)h αj(n−1)h Y(n−1)h
(Ynh
−Y(n−1)h
).
155
6 Comparison of the Continuous-Time and the Discrete-Time Problems
Similar as in (I) it follows that this converges in ucp to
Zt
αis αjs d
0
(Wsk + Aks )(Wsl + Als )
+
Zt
0
+
Zt
αis αjs (Wsk + Aks )dWsl
(6.14)
0
αis αjs (Wsk + Aks )dAls −
Zt
αis αjs (Wsl + Als )dWsk +
0
Zt
αis αjs (Wsl + Als )dAks
0
as h → 0. Whereas for l , k the sum (6.14) is zero by virtue of the independence of Wk and
Rt
Wl , for l = k the sum (6.14) is equal to αis αjs ds. Thus, the second sum tends in ucp for
0
h → 0 to
Zt
α⊤
s Σαs ds.
0
(III) It follows directly from the second sum that the third sum (III) converges in ucp to
d Z
X
t
αis
i=1 0
d
X
2
σik
ds
k=1
(IV) The process c̃ is càglàd and therefore Riemann integrable. Hence, the fourth sum (IV) tends
Rt
pointwise to c̃s ds as h → 0.
0
(V) For the last sum we consider
⌊ ht ⌋
X
n=1
|Z̃ni Z̃nj Z̃nk | ≤
⌊t⌋
d X
h
X
k=1 n=1
|Z̃nk |3
≤
d
X
sup
t
k=1 1≤n≤⌊ h ⌋
|Z̃nk |
for i, j, k ∈ {1, . . . , d} and t ∈ [0, T ]. From (III) we know that
to
d
Rt P
0 j=1
2 ds as h → 0. Moreover,
σkj
sup
1≤n≤⌊ ht ⌋
⌊ ht ⌋
X
l=1
⌊ ht ⌋
P
n=1
|Z̃lk |2 .
|Z̃nk |2 converges in ucp
|Z̃nk | tends to zero a.s. by the a.s. uniform
continuous paths of Yt over [0, T ]. Thus, the remainder r3n converges in ucp to zero as h → 0.
As convergence in probability is preserved under the sum the assertion follows from the considerations in (I)-(V), i.e. (6.13) converges in ucp to
Zt
0
α⊤
s (σ λ̂(s, IT , Ys ) − r1d ) −
1 ⊤
α Σαs − c̃s ds +
2 s
Zt
IT Y
.
α⊤
s σdWs
0
Eventually, we suppose π to be càdlàg. Then π − is càglàd. Hence, π h converges pointwise to π − .
Analogously to (I)-(V), we obtain the ucp-convergence of (6.12) to
Zt
0
156
(α− )⊤
s (σ λ̂(s, IT , Ys ) − r1d ) −
1
(α )⊤ Σ(α− )s − (c̃− )s ds +
2 − s
Zt
0
IT Y
.
(α− )⊤
s σdWs
(6.15)
6.2 Convergence Results
However, due to the continuity of the integrators
Zt
α⊤
s (σ λ̂(s, IT , Ys ) − r1d ) −
0
1 ⊤
α Σαs − c̃s ds +
2 s
Zt
IT Y
α⊤
s σdWs
0
is well-defined and coincides with (6.15).
Theorem 6.2.9. Let π = (c̃t , αt )t∈[0,T ] be given as in (6.11). Then
⌊T
h⌋
X
h
h γ
)γ h + (XN
)
(c̃(k−1)h Xk−1
k=1
−→
ZT
π
π
(c̃t Xt )γ dt + (XT )γ
0
in probability as h → 0.
Proof. Since convergence in probability is preserved under continuous maps it follows from Lemma
h )γ tends in ucp to (X π )γ as h → 0. Let us consider the first sum
6.2.7 that (XN
T
ZT
0
=xγ0
π
(c̃t Xt )γ dt −
ZT
0

c̃γt exp γ
⌊T
h⌋
X
h
(c̃(k−1)h Xk−1
)γ h
(6.16)
k=1
Zt
0
1 ⊤
r + α⊤
s (σ λ̂(s, IT , Ys ) − r1d ) − αs Σαs − c̃s ds +
2
⌊T
h⌋
Zt
0

IT Y 
dt
α⊤
s σdWs
!
k−1
X
X
−rh
log (1 − c̃(l−1)h h) 1 + α⊤
er(k−1)hγ (c̃(k−1)h )γ exp γ
− xγ0
R(Zlh ) − 1d )
.
(l−1)h (e
l=1
k=1
In order to simplify the notation we make use of the following abbreviations
Nt := γ
Zt
0
Mkh := γ
r + α⊤
s (σ λ̂(s, IT , Ys ) − r1d ) −
k X
l=1
1 ⊤
α Σαs − c̃s ds +
2 s
Zt
IT Y
α⊤
s σdWs
0
−rh
h
rh + log (1 − c̃(l−1)h h) 1 + α⊤
(e
R(Z
)
−
1
)
.
d
l
(l−1)h
We then decompose (6.16) in the following way
ZT
0
=xγ0
π
(c̃t Xt )γ dt −
ZT
0
⌊T
h⌋
X
h
(c̃(k−1)h Xk−1
)γ h
k=1
c̃γt exp (Nt ) dt − xγ0
⌊T
h⌋
X
c̃γ(k−1)h h exp N(k−1)h
k=1
⌊T
h⌋
⌊T ⌋
k=1
k=1
h
X γ
X
h
+ xγ0
c̃(k−1)h h exp N(k−1)h − xγ0
c̃γ(k−1)h h exp Mk−1
.
(I)
(II)
157
6 Comparison of the Continuous-Time and the Discrete-Time Problems
Due to the continuity of the stochastic integral and the left continuity of c̃ the first sum (I)
converges pointwise to zero (Riemann sums) as h → 0. Let us consider the second sum (II)
xγ0
⌊T
h⌋
h
c̃γ(k−1)h h exp N(k−1)h − exp Mk−1
X
k=1
T
h⌋
⌊X
c̃γ(k−1)h h ≤ T xγ0 sup exp (Ns ) − exp M⌊hs ⌋ .
≤ sup exp (Ns ) − exp M⌊hs ⌋ xγ0
h
0≤s≤T
h
0≤s≤T
k=1
Since convergence in probability is preserved under continuous maps the second sum (II) tends to
zero in probability as h → 0 by Lemma 6.2.7. Using that convergence in probability is preserved
under the sum completes the proof.
h∗
The optimal invested and consumed fractions (c̃h∗
n , αn ) of the h-investor are Markovian and
independent of the level of wealth (see Theorem 4.5.3), i.e.
h∗
h
c̃h∗
n = c̃n (IT , Yn )
and
h∗
h∗
αn
= αn
(IT , Ynh ).
The optimal continuous-time strategy without short selling constraints has the same properties.
Even though we do not know the optimal strategy π ∗ of (P̄ c ) it is somehow reasonable to suppose
that π ∗ has also these features. Under this assumption we can construct an admissible strategy
for the h-investor via π ∗ = (c̃∗t , α∗t )t∈[0,T ] . The process c̃∗ denotes the optimal fraction of wealth
which is consumed. Then we set
c̃hn := c̃∗nh ;
h
αn
:= α∗nh
for n = 0, . . . , N − 1. Since π ∗ is Markovian and independent of the wealth (by assumption),
π ∗t is at time points t = nh measurable with respect to σ(IT , Yt ) = σ(IT , Ynh ). Hence, π h =
h)
(c̃hn , αn
n∈{0,...,N } is an admissible strategy for the h-investor and is at time points t = nh identical
to the optimal strategy of (P̄ c ).
Corollary 6.2.10. Suppose that the optimal strategy π ∗ of (P̄ c ) is càglàd or càdlàg, Markovian
and the optimal consumed and invested fractions of the wealth do not depend on the level of wealth.
Then it holds:
h (x , i ) = V̄ ∗ (x , i ) = V̄ ∗ (x , i );
a) lim V0π
0 T
0π
0 T
0 T
h
0
h→0
b) lim V0h∗ (x0 , iT ) = V̄0∗ (x0 , iT );
h→0
h (x , i )| = 0
c) lim |V0h∗ (x0 , iT ) − V0π
0 T
h
h→0
for x0 ≥ 0 and iT ∈ I.
Proof. Under the given assumptions it holds by Theorem 6.2.9 that
⌊T
h⌋
X
k=1
158
πh h γ
πh h γ
(c̃hk−1 Xk−1
) h + (XN
)
−→
ZT
0
π∗
π∗
(c̃∗t Xt )γ dt + (XT )γ
6.2 Convergence Results
in probability as h → 0. Thus, we obtain by Fatou’s lemma for convergence in probability (see
Lemma A.3.1) that
 T

Z
∗
∗
π
π
V̄0∗ (x0 , iT ) = V̄0π∗ (x0 , iT ) = E  (c̃∗t Xt )γ dt + (XT )γ | IT = iT 
0

⌊T
h⌋
≤ lim inf E 
h→0
h
X
k=1
h
h

π h γ
π h γ
h
(c̃hk−1 Xk−1
) h + (XN
) | IT = iT  = lim inf V0π
h (x0 , iT ).
h→0
where Xnπ h is the wealth process of the h-investor under strategy π h (the first equality holds under
h (x , i ) ≤
the given assumptions by definition). On the other hand, we have the inclusion V0π
0 T
h
h∗
∗
V0 (x0 , iT ) ≤ V̄0π∗ (x0 , iT ) = V̄0 (x0 , iT ). Hence,
lim V h h (x0 , iT ) = lim V0h∗ (x0 , iT ) = V̄0∗ (x0 , iT ).
h→0 0π
h→0
6.2.3 Conclusion
The main concern of this work was to show how consumption-investment problems in a discretized
Black-Scholes market with partial and insider information can be solved. For logarithmic and
power utility functions we derived optimal consumption-investment strategies for an investor
with partial and insider information who can trade and observe the financial market only on a
equidistant time grid with mesh size h > 0. Nevertheless, we were not able to present optimal
consumption-investment strategies in explicit form. That is why, we firstly provided a numerical
example and secondly also investigated the continuous-time consumption-investment problem.
Here we derived closed form solutions. Moreover, we proved that the continuous-time investor and
the h-investor use the same estimator for the drift (with insider information). However, in general
the expected utility of the h-investor does not converge to the expected utility of the continuoustime investor when h tends to zero. Hence, the h-investor faces a so-called discretization gap,
i.e.
lim |V0∗ (x0 , iT ) − V0h∗ (x0 , iT )| > 0.
h→0
If we have only insider information (the investor knows the drift), then the discretization gap can
also not be avoided. The main reason for the discretization gap is that the h-investor’s optimal
strategy excludes short selling whereas for the continuous-time investor short selling might be
optimal. Consequently, the continuous-time strategy is not robust with respect to discretization.
Introducing short selling constraints on the continuous-time consumption-investment problem
overcomes the lack of robustness. It turned out that the discretization gap is just the difference
between the value of the continuous-time consumption-investment problems with and without
short selling constraints (compare Corollary 6.2.4 and Corollary 6.2.10).
Based on the solution of the continuous-time problem with short selling constraints, we were
able to construct a simple and good discrete-time consumption-investment strategy for the hinvestor with logarithmic utility functions. The constructed discrete-time strategy ensures that
the associated expected logarithmic utility of the h-investor converges to his optimal expected
logarithmic utility (see Theorem 6.2.3). This result is in line with the findings in Bäuerle et al.
[2012] where a terminal wealth problem (without consumption) of an h-investor with logarithmic
159
6 Comparison of the Continuous-Time and the Discrete-Time Problems
utility functions and only partial information (without insider information) was investigated.
So far, we were not able to extend latter result to the more general power utility function, i.e. we
could not derive a simple closed form strategy which is a good approximation of the h-investor’s
optimal consumption-investment strategy with power utility functions.
160
A Appendix
A.1 Results for Chapter 5
Lemma A.1.1. It holds that
ZT X
d
0 k=1
2
E[ Kt−1 (ax − (a2 λT + a2 Wt )) k M x (t, Wt ) ]dt < ∞
[Used in the proof of Theorem 5.2.2].
Proof. We plug in M x (t, Wt ) (see Theorem 5.2.2) and obtain
ZT X
d
=
0 k=1
ZT d
X
0 k=1
2
E[ Kt−1 (ax − (a2 λT + a2 Wt )) k M x (t, Wt ) ]dt
"
E
2
Kt−1 (ax − (a2 λT + a2 Wt )) k
#
det(A) −(x−(aλT +aWt ))⊤ K −1 (x−(aλT +aWt ))+(x−aλ0 T )⊤ A−1 (x−aλ0 T )
t
dt
e
×
det(Kt )
≤e
(x−aλ0 T )⊤ A−1 (x−aλ0 T )
ZT
0
d
det(A) X E
det(Kt )
k=1
Kt−1 (ax − (a2 λT + a2 Wt ))
|
{z
}
=:Y (t)
2 k
dt.
For the last inequality we use that (x − (aλT + aWt ))⊤ Kt−1 (x − (aλT + aWt )) ≥ 0 due to the
positive definiteness of Kt−1 . Since det(Kt ) , 0 for all a ∈ [0, 1) and t ∈ [0, T ] the entries of Kt−1
are continuous functions of the entries of Kt and therefore bounded on the compact interval [0, T ].
The continuity of the (kj)-th entry of Kt−1 can be easily seen from the following formula for the
entries of Kt−1
(Kt−1 )jk = (−1)j+k
det(K̃tjk )
,
det(Kt )
(A.1)
where K̃tjk denotes the matrix obtained by removing the j-th row and the k-th column (note that
(−1)j+k det(K̃tjk ) are the entries of the adjugate of K). Moreover, Yt is (multivariate) normally
distributed such that all moments exist. Hence
e
⊤ −1
1
(x−aλ0 T )
2 (x−aλ0 T ) A
ZT
0
d
2 det(A) X E Kt−1 (ax − (a2 λT + a2 Wt )) k dt < ∞.
det(Kt )
k=1
161
A Appendix
Lemma A.1.2. The filtering problem in Theorem 5.2.9 satisfies the conditions (1)-(10) of Theorem 12.7 in Liptser & Shiryaev [2001b].
Proof. We abbreviate the 2d-dimensional observation process by Ȳ c = (Ȳtc )t∈[0,T ] with Ȳtc :=
Yt
and introduce the notations of Theorem 12.7:
Itc
a0 (t, Ȳ c ) :=
0d×1 ,
a1 (t, Ȳ c )
A0 (t, Ȳ c ) :=
02d×1 ,
A1 (t, Ȳ c )
b1 (t, Ȳ c ) :=
0d×d ,
b2 (t, Ȳ c )
B1 (t, Ȳ c ) :=
02d×d ,
B2 (t, Ȳ c )
b ◦ b :=
B1 B1⊤ + B2 B2⊤
b ◦ B :=
b1 B1⊤ + b2 B2⊤
:= −a2 Kt−1 0d×d ;
Id
0d×d
:=
0d×d cId
= a4 Kt−1 Kt−1 ;
Id
0d×d
;
=
0d×d c2 Id
⊤
b1 b⊤
1 + b2 b2
B ◦ B :=
:= 0d×d ;
Id
;
:=
0d×d
0d×d .
= −a2 Kt−1
We check the conditions (1) − (10) (see pp. 32–35 in Liptser & Shiryaev [2001b]):
(1) X since:
Since det(Kt ) , 0 for all t ∈ [0, T ] the entries of Kt−1 are continuous functions of the entries
of Kt (compare (A.1)) and therefore bounded on the compact interval [0, T ]. Hence,
ZT X
2d d X
=
0 i=1 j=1
ZT X
d
4
a
0 i,j=1
−a2 Kt−1
Kt−1
2
ij
0d×d
2
ij
+
2d
X
i,j=1
Id
0d×d
0d×d
cId
!2
dt
ij
+ d + dc2 dt < ∞;
(2) X since:
The matrices A0 and A1 do not depend on t;
(3) X since:
Due to the modification with a constant c > 0 the matrix B ◦ B is non-singular with
Id
0d×d
−1
. Especially, (B ◦ B)−1 does not depend on t and Ȳ c .
(B ◦ B) =
0d×d c12 Id
(4) X since:
The matrices B1 , B2 do not depend on t, Ȳ c ;
(5) X since:
The random variable λ + ηtS is multivariate normally distributed with
Hence,
RT
0
162
d
λ + ηtS = N λ0 , Σ0 + a4 (T − t)Kt−1 Kt−1 + a2 (1 − a)2 Kt−1 Σε Kt−1 .
E[|λk + ηtSk |]dt < ∞, k = 1, . . . , d;
(A.2)
A.1 Results for Chapter 5
(6) X (compare (5));
(7) X since:
By Jensen’s inequality and by Fubini’s theorem it holds that
 T

Z
RT
c
c
E  (E[λk + ηtSk | FtI Y ])2 dt ≤
E[E[(λk + ηtSk )2 | FtI Y ]]dt < ∞
0
0
for k = 1, . . . , d (note that λk + ηtSk is normally distributed).
(8) X since:
The matrices A1 and a1 do not depend on t, Ȳ c ;
(9) X since:
All entries of Kt−1 are continuous on [0, T ] such that
(10) X since:
d
RT P
0 i,j=1
a4
Kt−1
2
ij
dt < ∞;
For k = 1, . . . , d the random variables λk + η0Sk are normally distributed. Hence,
η0Sk )4 < ∞, k = 1, . . . , d.
d
P
E(λk +
k=1
Lemma A.1.3. Let for s ≤ t the sets Hs,t and Ms be defined by
Hs,t := Z R-valued, bounded and Gs − measurable random variable :
E[Z(WtSk − WsSk )] = 0, k = 1, . . . , d ;
Ms :={1A∩C : A ∈ Fs and C ∈ σ(ITS )}.
Then Hs,t is a monotone vector space and Ms is a multiplicative class with σ(Ms ) = Gs .
Proof. Hs,t is a monotone vector space (see Definition in Protter [1990]):
(i) 1Ω ∈ Hs,t (Obviously by (5.4));
(ii) Hs,t is a vector space over R (Z1 , Z2 ∈ Hs,t , λ ∈ R ⇒ Z1 + Z2 ∈ Hs,t , λZ1 ∈ Hs,t );
(iii) Let (Zn )n∈N with Zn ∈ Hs,t be a non-negative and increasing sequence with lim Zn = Z
n→∞
(pointwise). Then, Z is bounded and Gs -measurable as the pointwise limit of Gs -measurable
random variables. Finally, by dominated convergence we obtain
E[ lim Zn (WtSk − WsSk )] = lim E[Zn (WtSk − WsSk )] = 0
n→∞
n→∞
Note that the sequence (Zn )n∈N is bounded such that there exists a c ≥ 0 with |Zn (WtSk −
WsSk )| ≤ c|WtSk − WsSk |.
Ms is multiplicative:
Let f, g ∈ Ms , i.e. f = 1A1 ∩C1 , A1 ∈ Fs , C1 ∈ σ(ITS ) and g = 1A2 ∩C2 , A2 ∈ Fs , C2 ∈ σ(ITS ). Since
A1 ∩ A2 ∈ Fs and C1 ∩ C2 ∈ σ(ITS ) it follows that f g ∈ Ms .
σ(Ms ) = Gs and Ms ⊆ Hs,t :
The first assertion follows from Lemma A.1.4. Let Zs ∈ Ms , i.e. Zs = 1A∩C , A ∈ Fs , C ∈ σ(ITS ).
Since σ(Ms ) = Gs we have Ms ⊆ Gs . Hence, Zs ∈ Gs . The random variable Zs is bounded and
finally it follows from (5.4) that Ms ⊆ Hs,t .
163
A Appendix
Lemma A.1.4. The following σ-fields are equivalent
σ({A ∩ C : A ∈ Fs , C ∈ σ(ITS )}) = σ({A ∪ C : A ∈ Fs , C ∈ σ(ITS )}).
Proof. Define
A := σ({A ∩ C : A ∈ Fs , C ∈ σ(ITS )}),
C := σ({A ∪ C : A ∈ Fs , C ∈ σ(ITS )}).
(A.3)
In order to show A = C it is enough to show: (i) {A ∩ C : A ∈ Fs , C ∈ σ(ITS )} ⊆ C and (ii) A ⊇
{A ∪ C : A ∈ Fs , C ∈ σ(ITS )}.
"⊆" Let B ∈ {A ∩ C : A ∈ Fs , C ∈ σ(ITS )} (generator of A), i.e. B = A ∩ C with A ∈ Fs , C ∈ σ(ITS ).
Since AC ∈ Fs and C C ∈ σ(ITS ) we have B C = (A ∩ C)C = AC ∪ C C ∈ {A ∪ C : A ∈ Fs , C ∈
σ(ITS )}. By virtue of {A ∪ C : A ∈ Fs , C ∈ σ(ITS )} ⊆ C it holds B C ∈ C. Hence B = (B C )C ∈ C,
i.e. {A ∩ C : A ∈ Fs , C ∈ σ(ITS )} ⊆ C and therefore A ⊆ C.
"⊇" Let B ∈ {A ∪ C : A ∈ Fs , C ∈ σ(ITS )} (generator of C), i.e. B = A ∪ C with A ∈ Fs , C ∈ σ(ITS ).
Since AC ∈ Fs and C C ∈ σ(ITS ) we have B C = (A ∪ C)C = AC ∩ C C ∈ {A ∩ C : A ∈ Fs , C ∈
σ(ITS )}. By virtue of {A ∩ C : A ∈ Fs , C ∈ σ(ITS )} ⊆ A it holds B C ∈ A. Hence, B = (B C )C ∈
A, i.e. {A ∪ C : A ∈ Fs , C ∈ σ(ITS )} ⊆ A and therefore C ⊆ A.
Remark A.1.5. Recall that Gs is defined as the smallest σ-field containing both Fs as well as
σ(ITS ) and is generated by sets of the form A ∪ C with A ∈ Fs and C ∈ σ(ITS ).
Lemma A.1.6. Let Assumption 5.3.2 be satisfied. The semimartingale L=(f (t, λ̂t (t, IT , Yt ))t∈[0,T ]
(w.r.t. FIT Y ), where
f (t, λ̂(t, IT , Yt ))
q
1
(λ̂(0, IT , 0) − rσ −1 1d )⊤ (ΣIaT )−1 (λ̂(0, IT , 0) − rσ −1 1d )
:= det(Id + ΣIaT t) exp
2
1
× exp − (λ̂(t, IT , Yt )−rσ −1 1d )⊤ (Id + ΣIaT t)(ΣIaT )−1 (λ̂(t, IT , Yt )−rσ −1 1d ) ,
2
satisfies the following SDE
dLt = −Lt λ̂(t, IT , Yt ) − rσ −1 1d dWtIT Y ,
L0 = 1.
(A.4)
Proof. We make use of the abbreviation λ̂t := λ̂(t, IT , Yt ) and apply Itô’s formula to f . Therefore,
we determine the relevant partial derivatives
1
1
ft (t, λ̂t ) = tr (Id + tΣIaT )−1 ΣIaT f (t, λ̂t ) − (λ̂t − rσ −1 1d )⊤ (λ̂t − rσ −1 1d )f (t, λ̂t )
2
2
fλ̂ (t, λ̂t ) = − f (t, λ̂t )(λ̂t − rσ −1 1d )⊤ (Id + ΣIaT t)(ΣIaT )−1
fλ̂λ̂ (t, λ̂t ) = − f (t, λ̂t )(Id + tΣIaT )(ΣIaT )−1
+ f (t, λ̂t )(ΣIaT )−1 (Id + tΣIaT )(λ̂t − rσ −1 1d )(λ̂t − rσ −1 1d )⊤ (Id + tΣIaT )(ΣIaT )−1 .
With d[λ̂· , λ̂· ]t = (Id + tΣIaT )−1 ΣIaT ΣIaT (Id + tΣIaT )−1 dt we obtain by Itô’s formula
df (t, λ̂t ) = −f (t, λ̂t )(λ̂t − rσ −1 1d )⊤ dWtIT Y .
Hence f (t, λ̂(t, IT , Yt )) with f (0, λ̂(0, IT , 0)) = 1 solves the SDE (A.4).
164
A.1 Results for Chapter 5
Lemma A.1.7. Let
q
1
IT
−1
⊤
IT −1
−1
(λ̂(0, IT , 0) − rσ 1d ) (Σa ) (λ̂(0, IT , 0) − rσ 1d )
LT := det(Id + Σa T ) exp
2
1
−1
⊤
IT
IT −1
−1
× exp − (λ̂(T, IT , YT )−rσ 1d ) (Id + Σa T )(Σa ) (λ̂(T, IT , YT )−rσ 1d ) .
2
Then E[LT | F0IT Y ] = 1.
Proof. Since λ̂(0, IT , 0) is measurable w.r.t. F0IT Y it holds
E[LT | F0IT Y ]
q
1
= det(Id + ΣIaT T ) exp
(λ̂(0, IT , 0) − rσ −1 1d )⊤ (ΣIaT )−1 (λ̂(0, IT , 0) − rσ −1 1d )
2
1
× E exp − (λ̂(T, IT , YT )−rσ −1 1d )⊤ (Id + ΣIaT T )(ΣIaT )−1 (λ̂(T, IT , YT )−rσ −1 1d ) | F0IT Y .
2
Therefore, we consider now only the expectation in the above equation. λ̂(T, IT , YT ) given F0IT Y
is normally distributed, more precisely
d
λ̂(T, IT , YT ) − rσ −1 1d = N (b, B)
IT Y
F0
with b := λ̂(0, IT , 0) − rσ −1 1d , B := T ΣIaT ΣITT and ΣITT := (Id + T ΣIaT )−1 ΣIaT . Then
1
E exp − (λ̂(T, IT , YT )−rσ −1 1d )⊤ (Id + ΣIaT T )(ΣIaT )−1 (λ̂(T, IT , YT )−rσ −1 1d ) | F0IT Y
2
!
Z
1
1
1 ⊤ IT −1
⊤ −1
=
exp − x (ΣT ) x − (x − b) B (x − b) dx
d p
2
2
(2π) 2 det(B)
s
!
det(((ΣITT )−1 + B −1 )−1 )
1 ⊤ −1
IT −1
−1
−1 −1 −1
exp − b B − B ((ΣT ) + B ) B
b
=
det(B)
2
Z
⊤
1
1
q
×
exp − (x − (ΣITT )−1 + B −1 )−1 B −1 b
d
2
(2π) 2 det(((ΣITT )−1 + B −1 )−1 )
!
IT −1
IT −1
−1
−1 −1 −1
× (ΣT ) + B
x − ((ΣT ) + B ) B b dx.
The last two lines become one since the integrand together with the square root can be seen
as the density of a multivariate normally distributed random variable. Summarizing the above
considerations we obtain
E[LT | F0IT Y ] = 1.
Lemma A.1.8. Let the process W Q = (WtQ )t∈[0,T ] be defined by
dWtQ = dWtIT Y + (λ̂(t, IT , Yt ) − rσ −1 1d )dt,
W0Q = 0.
Then W Q is a d-dimensional FIT Y -Brownian motion under Q.
165
A Appendix
Proof. We use here Lévy’s characterization of a Brownian motion.
(i) As the finite variation part has a quadratic variation of zero, it holds
[W·Q , W·Q ]t = [W·IT Y , W·IT Y ]t = tId .
(ii) According to Hunt & Kennedy [2004] Lemma 5.19 W Q is an FIT Y -local martingale under
Q if and only if (LW Q ) is an FIT Y -local martingale under P. Hence, for k = 1, . . . , d we
consider (Lt WtQk ) and apply Itô’s formula
d(Lt WtQk ) =WtQk dLt + Lt dWtQk + d[W·Qk , L· ]t
= − WtQk Lt λ̂(t, IT , Yt ) − rσ −1 1d dWtIT Y + Lt dWtIT Y k
+ Lt (λ̂(t, IT , Yt ) − rσ −1 1d )k dt − Lt (λ̂(t, IT , Yt ) − rσ −1 1d )k dt.
Since the drift term cancels out it follows that (LW Q ) is a local martingale under P and
thus W Q is a local martingale under Q.
By Protter [1990] Theorem 39 W Q is an FIT Y -Brownian motion under Q.
Lemma A.1.9. The measure Q is a martingale measure equivalent to P (on (Ω, FTIT Y )), i.e. the
discounted stock price processes are local martingales under Q with respect to FIT Y . Moreover,
the discounted wealth process admits the representation
π
Xπ
X0π
ct
Xt
= t αt⊤ σdWtQ −
dt,
= x0 .
(A.5)
d
Bt
Bt
Bt
B0
Proof. In the market with information filtration FIT Y the dynamics of the stock price processes
are given by
dStk = Stk
d
X
j=1
σkj (λ̂(t, IT , Yt ))j dt + dWtIT Y j ,
k = 1, . . . , d.
By Itô’s formula and by Lemma A.1.8 we obtain the dynamics of the discounted stock price
processes under Q


k
d
d
d
Stk X
Sk X
Sk X
St
=
(σkj (λ̂(t, IT , Yt ))j − r dt + t
σkj dWtIT Y j = t
σkj dWtQj .
d
Bt
Bt
Bt
Bt
j=1
j=1
j=1
Hence, the discounted stock price processes are local martingales since the drift term cancels
out under Q. In the same manner we obtain the discounted wealth process under Q as given in
(A.5).
A.2 Results for Chapter 6
Lemma A.2.1. Let cν = (cνt )t∈[0,T ] be a consumption strategy and C ≥ 0 an FTIT Y -measurable
random variable with


ZT
x0 = E HTν C + Hsν cνs ds|F0IT Y 
0
for any dual process ν ∈ H. Then there exists an investment strategy αν = (αtν )t∈[0,T ] such that
ν
ν
π ν = (cνt , αtν )t∈[0,T ] ∈ A0 with Xtνπ ≥ 0 and XTνπ = C.
166
A.2 Results for Chapter 6
Proof. We consider the process X = (Xt )t∈[0,T ] with


ZT
1  ν
Xt := ν E HT C + Hsν cνs ds | FtIT Y  .
Ht
t
Then Xt satisfies X0 = x0 and XT = C. Moreover, we define the process M = (Mt )t∈[0,T ] by
Mt :=

E HTν C +
ZT
Hsν cνs ds
0
| FtIT Y

=
Xtν Htν
+
Zt
Hsν cνs ds.
0
RT
Then M is an FIT Y -martingale closed by HTν C + Hsν cνs ds with sup E[|Mt |] = x0 . By Theorem
t∈[0,T ]
0
5.3.8 there exists an FIT Y -progressively measurable process φ = (φt )t∈[0,T ] with
a.s. such that
Mt = x0 +
Zt
RT
0
φ⊤
t φt dt < ∞
IT Y
φ⊤
.
s dWs
0
A comparison with (6.2) yields that X is the wealth process under strategy π ν = (cνt , αtν )t∈[0,T ]
with
φt
ν
−1
ν
αt = σ
+ θt .
Htν Xtνπ
d
P
Lemma A.2.2. The Taylor expansion of f (x1 , . . . , xd ) := log 1 +
αk (exk − 1) , where α =
k=1
(α1 , . . . , αd )⊤ ∈ D, around 0d×1 is given by
f (x1 , . . . , xd ) =
d
X
i=1
with |r3 | ≤
d
P
k,i,j=1
αi xi −
d
d
1 X
1X
αi αj xi xj +
αi x2i + r3
2
2
i,j=1
i=1
|xi xj xk |.
Proof. Expanding f in a Taylor series around 0 gives us
f (x) = f (0) +
d
X
i=1
d
1 X
∂
∂2
+
+ r3 .
xi
f (x)
f (x)
xi xj
∂xi
2
∂xi ∂xj
x=0
x=0
i,j=1
Using the Lagrange form we can write the remainder of third degree r3 as follows
r3 =
d
1 X
∂3
xi xj xk
f (x)
6
∂xi ∂xj ∂xk
x=ξx
i,j,k=1
167
A Appendix
for some ξ ∈ [0, 1]. We compute the partial derivatives needed in the Taylor expansion: Let
i, j, k ∈ {1, 2, 3} with i , j , k. Then
αi exi
∂
f (x) =
∂xi
1 + α⊤ (ex1 , . . . , exd )⊤ − 1
d

2
xi
∂2
α
e
αi exi
i
−

f (x) =
⊤
(∂xi )2
⊤
⊤
x
x
x
1
1
d
1 + α (e , . . . , e ) − 1
1 + α (e , . . . , exd )⊤ − 1
d
∂2
d
αi exi αj exj
f (x) = − 2
1 + α⊤ (ex1 , . . . , exd )⊤ − 1d

3 
2
x
x
3
i
i
αi e
αi e
∂
 −3 

f (x) =2 
⊤
(∂xi )3
⊤
⊤
x
x
x
1 + α (e 1 , . . . , e d ) −1d
1+α (e 1 , . . . , exd )⊤ −1d


xi
α
e
i

+
1 + α⊤ (ex1 , . . . , exd )⊤ − 1d
∂xi ∂xj
∂3
f (x) =− (∂xi )2 ∂xj
αi exi αj exj
1+α⊤ (ex1 , . . . , exd )⊤ − 1d
∂3
f (x) = − ∂xi (∂xj )2
∂3
f (x) =2 ∂xi ∂xj ∂xk
We observe that 0 ≤
αi exi αj exj
2 +2 1 + α⊤ (ex1 , . . . , exd )⊤ − 1d
αi2 e2xi αj exj
1+α⊤ (ex1 , . . . , exd )⊤ −1d
2 + 2 1 + α⊤ (ex1 , . . . , exd )⊤ − 1d
αi exi αj exj αk exk
3 .
1 + α⊤ (ex1 , . . . , exd )⊤ − 1d
αi exi
1+α⊤ ((ex1 ,...,exd )⊤ −1d )
αi exi αj2 e2xj
3
≤ 1 and therefore r3 ≤
follows by setting x = 0 in the partial derivatives.
d
P
i,j,k=1
3
|xi xj xk |. The assertion
Lemma A.2.3. For the investment strategy αh as given in Theorem 6.2.3 it holds:
h i
E log 1 + (αhn−1 )⊤ (e−rh R(Znh ) − 1d ) |IT
3
1
=E h(αhn−1 )⊤ (σ λ̂((n − 1)h, IT , Y(n−1)h ) − r1d ) − h(αhn−1 )⊤ Σαhn−1 |IT + o(h 2 ).
2
Proof. Let




exp(Z̃n1 )




..
f (Z̃n1 , . . . , Z̃nd ) := log 1 + (αhn−1 )⊤ 
 − 1d  .
.
exp(Z̃nd )
with Z̃nk := σk Znh − (r + 21 σk σk⊤ )h, k = 1, . . . , d, and σk := (σk1 , . . . , σkd ), k = 1, . . . , d, i.e. σk is the
k-th row of σ. Using the tower property of the conditional expectation and the Taylor expansion
168
A.2 Results for Chapter 6
of f around 0 as given in Lemma A.2.2 we obtain
i
h E log 1 + (αhn−1 )⊤ (e−rh R(Znh ) − 1d ) |IT
i
h
h
=E E[f (Z̃n1 , . . . , Z̃nd )|IT , Z1h , . . . , Zn−1
] IT
d
d
d
h hX
i i
1 X hi
1 X hi
hj
i
i j
h
Z̃
−
α
α
αn−1 (Z̃ni )2 + r3 IT , Z1h , . . . , Zn−1
Z̃
Z̃
+
=E E
αhi
IT
n−1 n
n−1 n−1 n n
|{z}
2
2
i,j=1
i=1
i=1
(iv)
|
{z
}
|
{z
}
|
{z
}
(i)
with |r3 | ≤
d
P
i,j,k=1
(iii)
(ii)
|Z̃ni Z̃nj Z̃nk |. We consider (i)-(iv) separately:
(i) Thanks to (6.1) we immediately obtain
i
h
1 2
h
h
h
⊤
h
⊤
σ λ̂((n − 1)h, IT , Y(n−1)h ) − (r1d + σ̃ )
E (αn−1 ) Z̃n IT , Z1 , . . . , Zn−1 = h(αn−1 )
2
with σ̃ 2 := σ1 σ1⊤ , . . . , σd σd⊤
⊤
.
(ii) With (i) we get
i
h
h
E (αhn−1 )⊤ Z̃n Z̃n⊤ αhn−1 IT , Z1h , . . . , Zn−1
=h(αhn−1 )⊤ Σ + hσ(Id + (n − 1)hΣIaT )−1 ΣIaT σ
1
1
+ hσ λ̂((n − 1)h, IT , Y(n−1)h )λ̂((n − 1)h, IT , Y(n−1)h )⊤ σ + h(r1d + σ̃ 2 )(r1d + σ̃ 2 )⊤
2
2
1
− hσ λ̂((n − 1)h, IT , Y(n−1)h )(r1d + σ̃ 2 )⊤
2
1 2
⊤
−h(r1d + σ̃ )λ̂((n − 1)h, IT , Y(n−1)h ) σ αhn−1
2
(iii) We obtain (iii) directly from (ii) by selecting only the diagonal elements of the matrix in
between the αhn−1 .
3
(iv) We show that E[r3 |IT ] ≤ h 2 ζ for an integrable and σ(IT )-measurable random variable ζ
which is independent of h and n. Since
" d
#
d
X
X
i j k
k 3
E[|Z̃nk |3 |IT ],
E[|Z̃n Z̃n Z̃n | | IT ] ≤ E
|Z̃n | 1{|Z̃ k |=max{|Z̃ 1 |,...,|Z̃ d |}} |IT ≤
n
n
n
k=1
k=1
there exists a constant c such that
E[|r3 | | IT ] ≤
d
X
i,j,k=1
E[|Z̃ni Z̃nj Z̃nk | | IT ] ≤ c
d
X
k=1
34
E[|Z̃nk |3 |IT ] ≤ c E[(Z̃nk )4 |IT ] ,
where the last inequality follows by Jensen’s inequality. By Lemma A.2.4 the conditional
distribution of Z̃nk given IT follows a normal distribution which is independent of n, namely
1
N (h (σk (CλIT λ0 + C IT IT ) − (r + σk σk⊤ )), h σk (Id + ΣIaT h)σk⊤ ).
{z
}
2
|
{z
} |
=:g
=:gµ (IT )
σ
169
A Appendix
Determining the fourth moment yields
E[(Z̃nk )4 |IT ] =h2 (h2 gµ (IT )4 + 6hgµ (IT )2 gσ + 3gσ2 ).
3
Hence, E[|r3 | | IT ] ≤ h 2 ζ for an integrable random variable ζ depending on IT but which
can be chosen independent of h and n.
Furthermore, we are able to find for all terms in (ii) and (iii) containing h2 such an integrable
3
3
(bounding) random variable ζ as in (iv). Thus, we collect all these terms in o(h 2 ) with o(h 2 ) ≤
3
3
h 2 ζ (note that o(h 2 ) depends on IT ). All in all, we have
i
h E log 1 + (αhn−1 )⊤ (e−rh R(Znh ) − 1d ) |IT
h
1
=E h(αhn−1 )⊤ σ λ̂((n − 1)h, IT , Y(n−1)h ) − (r1d + σ̃ 2 )
2
i
3
1
1
− h(αhn−1 )⊤ Σαhn−1 + h(αhn−1 )⊤ σ̃ 2 |IT + o(h 2 )
2
2
h
i
1
3
=E h(αhn−1 )⊤ σ λ̂((n − 1)h, IT , Y(n−1)h ) − r1d − h(αhn−1 )⊤ Σαhn−1 |IT + o(h 2 ).
2
Lemma A.2.4. The conditional distribution of Zn given IT follows a normal distribution
N h(CλIT λ0 + C IT IT ), h(Id + ΣIaT h) .
Proof. Using the tower property of the conditional distribution, i.e.
i
h h ⊤
h ⊤
i i
E eip Zn |IT = E E eip Zn |IT , Z1 , . . . , Zn−1 |IT , p ∈ Rd ,
the assertion follows using results from Section 4.3.3.
Lemma A.2.5. Let αh be the strategy defined in Theorem 6.2.3. Then it holds:
a)
lim
⌊T
h ⌋
P
3
h(αhk−1 )⊤ (σ λ̂((k − 1)h, IT , Y(k−1)h ) − r1d ) − 21 h(αhk−1 )⊤ Σαhk−1 + o(h 2 )
h→0 k=1
RT
1 ⊤
= α⊤
t (σ λ̂(t, IT , Yt ) − r1d ) − 2 π t Σαt dt
0
lim
b)
N
P
h→0 k=1
=
RT Rs
0 0
Proof.
h
k−1
P
j=1
a.s.;
3
h(αhj−1 )⊤ (σ λ̂((j − 1)h, IT , Y(j−1)h ) − r1d ) − 12 h(αhj−1 )⊤ Σαhj−1 + o(h 2 )
1 ⊤
α⊤
u (σ λ̂(u, IT , Yu ) − r1d ) − 2 αu Σαu duds a.s..
a) The sums
⌊T
h⌋
P
k=1
h(αhk−1 )⊤ (σ λ̂((k − 1)h, IT , Y(k−1)h ) − r1d )
and
⌊T
h⌋
P
k=1
h(αhk−1 )⊤ Σαhk−1
are the standard Riemann sum approximations which converge due to the continuity of
t 7→ π t and t 7→ λ̂(t, IT , Yt ), i.e.
170
A.3 Auxiliary Results from (Stochastic) Analysis
(i)

lim h
h→0

P
k=1
= lim h
h→0

⌊T
h⌋
(αhk−1 )⊤ (σ λ̂((k − 1)h, IT , Y(k−1)h ) − r1d )
⌊T
h⌋
P
k=1

(π (k−1)h )⊤ (σ λ̂((k − 1)h, IT , Y(k−1)h ) − r1d )
RT
= π⊤
t (σ λ̂(t, IT , Yt ) − r1d )dt a.s.;
0
(ii) lim
⌊T
h⌋
P
h→0 k=1
RT
h(αhk−1 )⊤ Σαhk−1 = π ⊤
t Σπ t dt a.s.;
0
3
(iii) The remainder o(h 2 ) is independent of k. Hence, lim
⌊T
h⌋
P
h→0k=1
1
3
o(h 2 ) = lim o(h 2 ) = 0 a.s.
h→0
b) Part b) follows analogously to part a) (we take the "Riemann sums of the Riemann sums").
Lemma A.2.6. For t = nh ∈ [0, T ], y ∈ Z and iT ∈ I let
Z
1
log 1 + α⊤ (e−rh R(z) − 1d ) Q̂Z
fh (α) :=
n (dz | iT , y)
h
1
f (α) :=α⊤ (σ λ̂(t, iT , y) − r1d ) − α⊤ Σα.
2
Then it holds:
a) The sequence of functions fh converges uniformly to f over D.
b) The functions fh and f are strictly concave over D.
Proof.
a) By Taylor expansion in Lemma A.2.2 we obtain for t = nh and for fixed y ∈ Z, iT ∈ I
(similar to the proof of Lemma A.2.3) that
1
1
fh (α) = α⊤ (σ λ̂(t, iT , y) − r1d ) + α⊤ Σα + o(h 2 ).
h
1
Since D is compact we can chose o(h 2 ) independent of α. Hence,
sup |fh (α) − f (α)| −→ 0
α∈D
as h → 0.
b) Since −Σ is negative definite f is strictly concave. For the strict concavity of fh we refer
to the considerations in (4.22).
A.3 Auxiliary Results from (Stochastic) Analysis
Lemma A.3.1 and Lemma A.3.2 are direct consequences of Fatou’s lemma and Fubini’s theorem,
respectively (see e.g. Loève [1977]).
171
A Appendix
Lemma A.3.1. (Fatou’s Lemma for Convergence in Probability) Let G ⊆ F be a σ-field
p
and let (Xn )n∈N be a sequence of non-negative and integrable random variables with Xn −→
p
X, n → ∞ (−→ denotes convergence in probability). Then it holds:
a) E[X] ≤ lim inf E[Xn ];
n−→∞
b) E[X | G] ≤ lim inf E[Xn | G].
n−→∞
Lemma A.3.2. (Fubini’s Theorem for Conditional Expectation) Let X = (Xt )t∈[0,T ] be an
Rt
F-adapted càglàd or càdlàg process with E[|Xu |]du < ∞ P-a.s. for s, t ∈ [0, T ], s ≤ t. Moreover,
let G ⊆ F be a σ-field. Then it holds:
Zt
s
s
 t

Z
E[Xu | G]du = E  Xu du | G  a.s.
s
Lemma A.3.3. Let fn : D → R, where D ⊂ Rd is a compact set, be a sequence of functions which
converges uniformly to f : D → R. Let fn and f be strictly concave and
∗
αn
:= arg max fn (α)
α∈D
and
α∗ := arg max f (α).
α∈D
∗ −→ α∗ as n → ∞.
Then αn
∗ are the unique maximum points of f and f , respectively, the following
Proof. Since α∗ and αn
n
inequalities hold
∗
fn (α∗ ) ≤ fn (αn
)
and
∗
f (αn
) ≤ f (α∗ ).
It follows
∗
∗
∗
fn (α∗ ) − f (α∗ ) ≤ fn (αn
) − f (α∗ ) ≤ fn (αn
) − f (αn
).
From the uniformly convergence of f we obtain
∗
∗
∗
)|, |fn (α∗ ) − f (α∗ )| } ≤ sup |fn (α) − f (α)| −→ 0 (A.6)
) − f (αn
) − f (α∗ )| ≤ max{ |fn (αn
|fn (αn
α∈D
for n → ∞. Let ε > 0. Then there exists an n0 ∈ N such that for all n ≥ n0
ε
sup |fn (α) − f (α)| <
2
α∈D
and by inequality (A.6) we get
(1)
(2)
∗) <
f (α∗ ) − fn (αn
∗ ) − f (α∗ ) <
fn (αn
n
ε
2
ε
2.
By adding (1) and (2) we obtain
∗
f (α∗ ) − f (αn
) < ε.
(A.7)
We choose an open environment U of α∗ (i.e. α∗ ∈ U ) and denote by U C its complement. Then
U C ∩ D is compact and due to the continuity of f the maximum max f (α) exists. However,
α∈U C ∩D
the maximum of f on D is unique. Hence,
f (α∗ ) −
172
max
α∈U C ∩D
f (α) > 0.
A.4 Some Properties of Matrices used in Chapter 5
Now we choose ε = f (α∗ ) −
max
α∈U C ∩D
f (α). With (A.7) it then follows
∗
f (αn
)>
max
α∈U C ∩D
f (α).
∗ ∈ U. Since U can be chosen arbitrarily small the assertion follows.
Hence αn
A.4 Some Properties of Matrices used in Chapter 5
A proof of the following result can e.g. be found in Harville [2008] Section 15.8.
Lemma A.4.1. For t ∈ [0, T ] let Mt ∈ Rd×d be non-singular. Additionally, let the entries of Mt
be continuously differentiable with respect to t. Then it holds:
d
det(Mt ) = det(Mt )tr(Mt−1 dt
Mt ).
a)
d
dt
b)
−1
d
dt (Mt )
d
= −Mt−1 dt
(Mt )Mt−1 .
Lemma A.4.2. Let dIkT , k = 1, . . . , d, be the eigenvalues of ΣIaT . Moreover, let (Id + ΣIaT t) be
non-singular. Then it holds:
a) tr((Id + ΣIaT t)−1 ΣIaT ) =
b) tr
d
P
I
dkT
I
T
k=1 1+dk t
;
(Id + ΣIaT t)−1 ΣIaT (Id + ΣIaT t)−1 ΣIaT
=
d
P
k=1
I
dkT
I
1+dkT t
2
.
Proof. We make use of the abbreviation D := diag(dI1T , . . . , dIdT ). Then ΣIaT = ODO⊤ for an
orthogonal matrix O.
a)
tr((Id + ΣIaT t)−1 ΣIaT ) =tr O⊤ (Id + ODO⊤ t)−1 OD = tr (Id + Dt)−1 D


 1

 IT
0
IT
0
d
1+d
t
1
1






..
..
=tr 




.
.



1
I
T
0
0
dd
IT
1+dd t
d
X
dIkT
=
;
IT
k=1 1 + dk t
b)
tr (Id + ΣIaT t)−1 ΣIaT (Id + ΣIaT t)−1 ΣIaT
=tr (Id + ODO⊤ t)−1 ODO⊤ (Id + ODO⊤ t)−1 ODO⊤
!2
d
X
dIkT
=
.
IT
k=1 1 + dk t
173
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177
Zusammenfassung
Ein zentrales Problem der Finanzmathematik sind Konsum-Investitionsprobleme. Das Hauptaugenmerk liegt dabei auf der Bestimmung einer optimalen Konsum-Investitionsstrategie. Ziel ist,
diejenige Konsum-Investitionsstrategie zu bestimmen, die bei gegebenem Anfangsvermögen und
festgelegtem Planungshorizont den Nutzen aus zwischenzeitlichem Konsum und Endvermögen
maximiert.
Historisch gesehen geht das Konsum-Investitionsproblem mit dem maximal erwarteten Nutzen
als Optimalitätskriterium auf Robert C. Merton zurück (siehe Merton [1969, 1971]). Im Rahmen
eines klassischen Black-Scholes-Marktes fand er für Log- und Power-Nutzenfunktionen heraus,
dass es optimal ist, stets einen konstanten Anteil am Vermögen in die risikobehaftete Anlagen
zu investieren und die Konsumrate proportional zum gegenwärtigen Vermögen zu halten. Noch
heute erfreut sich das Optimalitätskriterium von Merton kombiniert mit einem mehrdimensionalen Black-Scholes-Markt großer Beliebtheit. Dort kann ein Investor sein Vermögen auf mehrere
risikobehaftete Anlagen (Aktien) und eine risikolose Anlage mit konstanter Zinsrate (Bond)
verteilen. Die Preisprozesse der risikobehafteten Anlagen folgen einer geometrisch Brown’schen
Bewegung. Die Annahme dabei, dass ein Investor die Marktparameter kennt, ist eher von theoretischer Natur. Vielmehr müssen diese aus den öffentlich zur Verfügung stehenden Marktdaten
geschätzt werden. Der Investor verfügt also nur über partielle Informationen. Während sich
die Volatilität aus den logarithmischen Preisen der risikobehafteten Anlagen bestimmen lässt,
ist die Schätzung der Drift deutlich schwieriger. Häufig wird diese unbekannte Drift daher als
eine Zufallsvariable mit einer gegebenen Anfangsverteilung modelliert, ein Bayes’scher Ansatz.
Einfach ausgedrückt: Der Investor zieht zufällig einen Wert für die Drift und passt diesen im
Laufe der Zeit entsprechend seiner Informationen an. Zu Portfoliooptimierungsproblemen mit
partiellen Informationen kombiniert mit dem Bayes’schen Ansatz gibt es eine umfangreiche Literatur, beispielsweise Lakner [1995] und Karatzas & Zhao [2001]. Allgemeinere Modelle mit unbekannter zeitabhängiger Drift werden unter anderem in Björk et al. [2010] und in Bäuerle &
Rieder [2005] betrachtet.
Insbesondere, wenn die Drift geschätzt werden muss, sieht sich mancher Investor auch nach externen Informationsquellen um. Der Öffentlichkeit nicht zur Verfügung stehende Informationen,
auch Insiderinformationen oder private Informationen genannt, können einem Investor Wettbewerbsvorteile gegenüber anderen Marktteilnehmern verschaffen. Einer der bekanntesten Fälle
von Insiderhandel in Deutschland spielte sich im Jahr 1993 ab. F. Steinkühler, damals Vorsitzender der IG Metall und Aufsichtsratsmitglied der Daimler-Benz AG, erwarb kurz vor der öffentlichen Bekanntgabe der geplanten Verschmelzung der Mercedes-Holding mit der Daimler-Benz
AG Mercedes-Aktien. Die Bekanntgabe führte zu einem deutlichen Kursanstieg.1 Inzwischen ist
Insiderhandel gesetzlich verboten, jedoch selten einfach nachzuweisen. Mathematische Modelle
für Insiderinformationen basieren häufig auf der Theorie der Vergrößerung von Filtrationen. Ausgangspunkt ist eine Filtration F. Diese soll alle öffentlich bekannten Informationen widerspiegeln.
Ein Insider hat jedoch Zugang zu einer Filtration G, die F enthält. Werden die Insiderinformationen zum Zeitpunkt Null hinzugefügt, spricht man von einer anfänglichen Vergrößerung der
Filtration. Die Theorie der anfänglichen Vergrößerung von Filtrationen geht ursprünglich auf Itô
1 vgl.
DER SPIEGEL 21/1993, Hamburg
179
Zusammenfassung
zurück und wurde von Jacod, Jeulin, Mansuy und Yor bedeutend weiterentwickelt. Die Arbeit
von Pikovsky & Karatzas [1996] ist eine der ersten, die Techniken der anfänglichen Vergrößerung
von Filtrationen verwendet, um den Einfluss von Insiderinformationen auf Konsum- und Investitionsentscheidungen zu untersuchen. In Danilova et al. [2010] werden partielle und private Informationen im Rahmen eines Portfoliooptimierungsproblems (ohne Konsum) kombiniert und
mit der Martingalmethode gelöst (dualer Ansatz). Im Gegensatz zu Danilova et al. [2010] löst
Hansen [2013] das Konsum-Investitionsproblem eines Investors mit Insiderinformationen mit Hilfe
der dynamischen Programmierung.
Die konstante Strategie im klassischen Merton-Modell ist etwas trügerisch. Denn um die Anteile konstant zu halten, muss aufgrund der sich ständig ändernden Aktienpreise der Investor
kontinuierlich handeln. In der Praxis ist kontinuierliches Handeln nicht möglich. Es gibt stets
Zeitspannen, in denen der Investor den Finanzmarkt nicht beobachtet und auch nicht handelt,
sei es um Transaktionskosten zu vermeiden oder nur um etwas zu relaxen. Ein h-Investor oder
relaxter Investor ist ein zeitdiskreter Investor, der nur zu Vielfachen von h > 0 den Finanzmarkt
beobachten und Konsum-Investitionsentscheidungen treffen kann. In einem Zeitintervall h können
sich die Aktienpreise derart ändern, dass Short-Selling bei einem h-Investor schnell zum Bankrott
führen kann und somit nicht optimal ist. Portfoliooptimierungsprobleme eines h-Investors mit
nur partiellen Informationen werden in Rogers [2001] und in Bäuerle et al. [2012] betrachtet.
In der vorliegenden Arbeit kombinieren wir diese verschiedenen Forschungsbereiche der Portfoliooptimierung: Konsum-Investitionsprobleme mit partiellen und privaten Informationen, und
eine Approximation des mehrdimensionalen Black-Scholes-Marktes mit zeitdiskretem Optimierungsproblem. In diesem Zusammenhang stellen sich einige Fragen:
• Ist es möglich, eine optimale Konsum-Investitionsstrategie eines Insiders in expliziter Form
zu bestimmen?
• Ist das Portfoliooptimierungsproblem ohne Konsum einfacher zu lösen?
• Hängt eine optimale Konsum-Investitionsstrategie von den Insiderinformationen ab?
• Welchen Wert haben Insiderinformationen für einen Investor?
• Hat der zeitstetige Insider einen Informationsvorteil gegenüber dem h-Insider?
• Konvergiert der Wert des zeitdiskreten Konsum-Investitionsproblems gegen den Wert des
zeitstetigen, wenn der Zeitabstand h gegen Null geht?
Ziel dieser Arbeit ist, eine optimale Konsum-Investitionsstrategie für einen h-Investor mit Logund Power-Nutzenfunktionen sowie partiellen und privaten Informationen herzuleiten. Im Vergleich zur bisher vorhandenen Literatur (z.B. Bäuerle et al. [2012]) werden hier partielle mit
privaten Informationen kombiniert. Da es aber schwierig ist eine optimal Strategie in geschlossener
Form anzugeben, betrachten wir auch das zeitstetige Konsum-Investitionsproblem (vgl. auch
Hansen [2013]) und leiten dafür eine optimale Lösung in expliziter Form her. Ausgehend von der
optimalen zeitstetigen Strategie (mit Short-Selling-Restriktionen) konstruieren wir (für kleine h)
eine gute Strategie für den h-Investor.
Diese Arbeit gliedert sich in weitere fünf Kapitel:
In Kapitel 2 und Kapitel 3 definieren wir das zugrundeliegende zeitstetige Modell: ein mehrdimensionaler Black-Scholes-Markt mit unbekannter Drift sowie Insiderinformationen. Die Volatilitätsmatrix der Aktien ist dem Investor bekannt und ohne Beschränkung der Allgemeinheit kann
180
diese im betrachteten Modell als symmetrisch und nichtsingulär angenommen werden. Die unbekannte Drift betreffend folgen wir einem Bayes’schen Ansatz mit einer multivariaten Normalverteilung als A-priori-Verteilung. Die Insiderinformationen werden als eine Zufallsvariable
modelliert, deren Wert ein Insider zu Beginn seiner Handelszeit kennt. Die Zufallsvariable ist
definiert als eine Konvexkombination bestehend aus einer unabhängigen Störung und den exakten Insiderinformationen. Diese sind entweder die terminalen Aktienpreise, die Drift oder der
terminale Wert der mehrdimensionalen Brown’schen Bewegung, die den Aktienpreisen zugrunde
liegt. In der Arbeit von Hansen [2013] werden exakte Informationen über die terminalen Aktienpreise oder die Drift durch eine additive Störung beeinträchtigt. Im Gegensatz zu Hansen
[2013] können wir den Skalierungsfaktor der Konvexkombination auf Null setzen, um das KonsumInvestitionsproblem mit nur partiellen Informationen zu erhalten. Im Rahmen eines eindimensionalen Black-Scholes-Marktes werden solche Konvexkombinationen in Danilova et al. [2010]
untersucht.
In Kapitel 4 untersuchen wir das Konsum-Investitionsproblem des h-Investors. Dazu diskretisieren wir den mehrdimensionalen Finanzmarkt und formulieren das Optimierungsproblem mathematisch. Mit Hilfe einer Filterrekursion (siehe Bäuerle & Rieder [2011]) nehmen wir in das
Optimierungsproblem die partiellen und privaten Informationen über eine Wahrscheinlichkeitsverteilung auf. Diese ist definiert als eine bedingte Verteilung der unbekannten Drift gegeben die verfügbaren Informationen und wird häufig als A-posteriori-Verteilung bezeichnet (siehe Satz 4.3.13).
Durch diesen Vergrößerungs- und Filterschritt können wir das Konsum-Investitionsproblem mit
der Theorie der Markov’schen Entscheidungsprozesse lösen. Dazu definieren wir einen Markov’schen Entscheidungsprozess mit vergrößertem Zustandsraum bestehend aus dem Vermögen, den
Insiderinformationen und der Beobachtung. Diese Beobachtung ist eine suffiziente Statistik für die
A-posteriori-Verteilung und hängt von den vergangenen Aktienpreisen ab. Damit lassen sich mit
der Bellman-Gleichung die Hauptresultate dieses Kapitels formulieren: in Satz 4.5.1 mit logarithmischen Nutzenfunktionen und in Satz 4.5.3 mit Power-Nutzenfunktionen. Jedoch können wir die
optimalen Strategien nicht in geschlossener Form angeben. Bei logarithmischen Nutzenfunktionen hängt die optimale Investitionsstrategie von den Insiderinformationen und der Beobachtung
ab, der optimale konsumierte Anteil am Vermögen dagegen nur von der Zeit. Des Weiteren hat
dieser konsumierte Anteil keinen Einfluss auf die Investitionsentscheidung des h-Investors, sodass das Portfoliooptimierungsproblem (ohne Konsum) und das Konsum-Investitionsproblem zu
der gleichen optimalen Investitionsstrategie führen. Das Konsum-Investitionsproblem mit PowerNutzenfunktion stellt sich als etwas komplizierter heraus. Hier hängt auch der optimale konsumierte Anteil am Vermögen von den Insiderinformationen und der Beobachtung ab. Um die drei
verschiedenen Insiderinformationen (Aktienpreise, Brown’sche Bewegung, Drift) zu vergleichen,
definieren wir den Wert der Insiderinformationen als ein Sicherheitsäquivalent. Es ist jedoch
schwierig, Eigenschaften dieses Sicherheitsäquivalents analytisch herzuleiten. Für eine bestimmte
Wahl der Parameter legt ein numerisches Beispiel allerdings die Vermutung nahe, dass ein hInvestor Informationen über die terminalen Aktienpreise gegenüber Informationen über die Drift
oder den terminalen Wert der Brown’schen Bewegung bevorzugt. Im Modell mit nur einer Aktie diskutieren wir die Monotonieeigenschaften der optimalen Strategien und der Wertfunktionen
bezüglich der Insiderinformation und der Beobachtung. Überraschenderweise ist die optimale
Strategie im Allgemeinen nicht monoton wachsend in der vom Aktienpreis abhängenden Beobachtung.
Zeitdiskrete Konsum-Investitionsprobleme mit partiellen und privaten Informationen wurden
bisher wenig untersucht. Unsere wichtigste Quelle hierzu ist das Buch von Bäuerle & Rieder
[2011].
In Kapitel 5 lösen wir das zeitstetige Konsum-Investitionsproblem. Im Vergrößerungs- und Filterschritt überführen wir mit Hilfe der Filtertheorie und mit Techniken der Vergrößerung von
181
Zusammenfassung
Filtrationen das Konsum-Investitionsproblem mit partiellen und privaten Informationen in eines
mit vollständigen Informationen. Analog zum h-Investor besteht der Zustandsprozess des zeitstetigen Investors dann aus dem Vermögensprozess, den Insiderinformationen und der aktuellen
Beobachtung der Aktienpreise. Mit der Hamilton-Jacobi-Bellman-Gleichung lässt sich die optimale Konsum-Investitionsstrategie explizit bestimmen: in Satz 5.5.2 mit logarithmischen Nutzenfunktionen und in Satz 5.5.6 mit Power-Nutzenfunktionen. Bei logarithmischen Nutzenfunktionen
hängt die optimale Investitionsstrategie linear von den Insiderinformationen und der Beobachtung
(der Aktienpreise) ab. Zudem gilt das "certainty equivalence principle", d.h. wir erhalten die optimale Investitionsstrategie, indem wir im Merton-Anteil die konstante Drift durch einen Schätzer
ersetzen, der aus dem Vergrößerungs- und Filterschritt folgt. Der optimale konsumierte Anteil ist hingegen nur eine zeitabhängige deterministische Funktion. Bei Power-Nutzenfunktionen
hat die optimale Konsum-Investitionsstrategie eine weitaus komplexere Gestalt. Betrachten wir
jedoch das Portfoliooptimierungsproblem ohne Konsum, vereinfacht sich die Investitionsstrategie beachtlich. Auch Hansen [2013] stellt eine Hamilton-Jacobi-Bellman-Gleichung auf, um das
Konsum-Investitionsproblem eines Insiders zu lösen. Er geht jedoch von einem alternativen Zustandsprozess aus, der aus dem Vermögen und dem Schätzer für die Drift besteht.
Am Ende des Kapitels wird der Wert der Insiderinformationen wieder als Sicherheitsäquivalent
definiert. Für einige Parametrisierungen sehen wir in Satz 5.6.5, dass ein Investor mit logarithmischen Nutzenfunktionen zusätzliche Informationen über den terminalen Aktienpreis gegenüber
zusätzlichen Informationen über die Drift oder den terminalen Wert der Brown’schen Bewegung
bevorzugt. Im Fall mit Power-Nutzenfunktionen ist die Gestalt des Sicherheitsäquivalents weitaus
komplizierter, es sei denn wir schließen Konsum aus (Korollar 5.6.8). In den Arbeiten von
Amendinger et al. [1998, 2003], Hansen [2013] und Liu et al. [2010] wird ein Sicherheitsäquivalent genauer untersucht. Anhand eines numerischen Beispiels veranschaulicht Hansen [2013]
seine theoretischen Resultate.
In Kapitel 6 vergleichen wir noch das zeitdiskrete und das zeitstetige Konsum-Investitionsproblem. Überraschenderweise verliert der relaxte Investor im Vergleich zum zeitstetigen Investor
keine relevanten Informationen: Zu den Handelszeitpunkten ergeben sich die gleichen Schätzer
für die Drift.
Wir betrachten zunächst das Konsum-Investitionsproblem mit Short-Selling-Restriktionen. Es
gibt zahlreiche Arbeiten, die Portfoliooptimierungsprobleme mit konvexen Beschränkungen behandeln. Unsere Hauptquelle hierzu ist Cvitanić & Karatzas [1992]. Dort wird ein fiktives
Finanzmarktmodell ohne Restriktionen eingeführt, mit dessen Hilfe sich das Konsum-Investitionsproblem mit konvexen Beschränkungen über den dualen Ansatz lösen lässt. Mit Resultaten von
Cvitanić & Karatzas [1992] und Dualitätseigenschaften der konvexen Optimierung zeigen wir in
Satz 6.2.1, dass sich die optimale Strategie über ein punktweises und quadratisches Maximierungsproblem bestimmen lässt. Durch dieses Optimierungsproblem mit Short-Selling-Restriktionen
sind wir in der Lage, einige Konvergenzeigenschaften des Konsum-Investitionsproblems eines hInvestors mit logarithmischen Nutzenfunktionen für h gegen Null aufzuzeigen. Im Allgemeinen
konvergiert der erwartete logarithmische Nutzen des zeitdiskreten Investors nicht gegen den des
zeitstetigen. Der Hauptgrund hierfür ist, dass Short-Selling für einen h-Investor nicht optimal
ist. Jedoch ist die diskretisierte optimale Strategie des zeitstetigen Investors mit Short-SellingRestriktionen für kleine h eine gute Strategie für den h-Investor (Satz 6.2.5). Für ein Portfoliooptimierungsproblem (ohne Konsum) mit nur partiellen Informationen (also ohne Insiderinformationen) wurden diese Resultate in Bäuerle et al. [2012] gezeigt. Zum jetzigen Zeitpunkt können
wir diese Ergebnisse nicht auf die allgemeinere Power-Nutzenfunktion ausweiten. Dennoch zeigen
wir unter gewissen Voraussetzungen und Short-Selling-Verbot, dass der erwartete Power-Nutzen
des h-Investors gegen den erwarteten Power-Nutzen des zeitstetigen Investors konvergiert (Satz
6.2.9 und Korollar 6.2.10).
182
Acknowledgements
First and foremost, I would like to express my deepest gratitude to my supervisor Prof. Dr. Ulrich
Rieder. He gave me the opportunity to write this thesis, supported my research, taught me how
to question thoughts and told me how to express my ideas. I really appreciate the many fruitful
discussions we had, and the valuable comments and suggestions he made during our almost weekly
meetings.
My very special thanks go to Prof. Dr. Robert Stelzer for not only being the co-examiner of
this thesis, but also for integrating me into the Institute of Mathematical Finance with various
off-campus activities.
I also wish to express thanks to all the other members of the Institute of Mathematical Finance.
Special thanks go to Dr. Zywilla Fechner and Dr. Imma Curato for proofreading parts of this
thesis and for many discussions not only about mathematics and economics.
Furthermore, I greatly appreciate the financial support of the Deutsche Forschungsgemeinschaft
as a member of the Research Training Group 1100. I always enjoyed being a member of this
research group. Without all my colleagues there, the great working atmosphere would not have
been possible.
Last and most importantly, I wish to thank my family - my parents, my brother and my sister for their constant love and support. Especially my parents encouraged and supported me in so
many ways from the very beginning of my studies until the end of this thesis.
Erklärung
Hiermit versichere ich, Julia Gentner, dass ich die vorliegende Arbeit selbständig angefertigt habe
und keine anderen als die angegebenen Quellen und Hilfsmittel benutzt sowie die wörtlich oder
inhaltlich übernommenen Stellen als solche kenntlich gemacht habe. Ich erkläre außerdem, dass
diese Arbeit weder im In- noch im Ausland in dieser oder ähnlicher Form in einem anderen
Promotionsverfahren vorgelegt wurde.
Ulm, den 17. Dezember 2014
(Julia Gentner)
Curriculum Vitae
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