CANADIAN APPLIED
MATHEMATICS QUARTERLY
Volume 17, Number 4, Winter 2009
TIME CONSISTENT UTILITY MAXIMIZATION
TRAIAN A. PIRVU AND ULRICH G. HAUSSMANN
ABSTRACT. This paper studies the problem of optimal
investment in incomplete markets, time consistent with respect
to stopping times investment horizons. We work in a Brownian
motion framework and the stopping times are adapted to the
Brownian filtration. Time consistency is achieved only if the
risk preferences are time and state dependent. Thus, they are
utilities random fields. Starting with a time and state independent utility we construct a utility random field which leads to
optimal time consistent investment. As an application to the
classical utility theory we provide a portfolio decomposition
formula.
1 Introduction Dynamic asset allocation has been an important
field in modern finance. The ground-breaking paper in this literature is
Merton [18]. He assumed a utility function of CRRA (constant coefficent
of relativ risk aversion) type and a market consisting of a risk-free asset
with constant rate of return and one or more stocks, each with constant
mean rate of return and volatility. Merton was able to derive closed
form solutions for the stochastic control problem of maximizing utility
of final wealth.
Karatzas et al. [12], and Cox and Huang [5] established the static
martingale method in solving the optimal investment and consumption
problem in the complete market paradigm. In incomplete markets, He
and Pearson [11] considered the minmax martingale measure to transform the dynamic portfolio allocation problem into a static one. An
auxiliary problem is analyzed, where the budget constraint is turned
into a static one using only one martingale measure (the minmax marWe thank Tahir Choulli and Ivar Ekeland for helpful discussions. Special thanks
are due to anonymous referees for theirs many suggestions that greatly helped to
improve the presentation of the results. Work supported by NSERC grants 37165309, 88051 and MITACS grants 5-26761, 30354.
AMS subject classification: (2000): 91B30, 60H30, 60G44.
Keywords: Portfolio optimization, incomplete markets, time consistency.
c
Copyright Applied
Mathematics Institute, University of Alberta.
721
722
T. A. PIRVU AND U. G. HAUSSMANN
tingale measure). Kramkov and Schachermayer [16] analysed the problem of maximizing utility from final wealth in a general semmimartingale
model by means of duality. Some works study the case where the investment horizon is random. This goes back to Levhari and Mirman [17].
One can regard the random time horizon as a major event (e.g., death
time) and by time change, it can be turned into a stochastic clock. Gol
and Kallsen [10] solved the problem of logarithmic utility maximization
in a general semimartingale framework. In a similar setup Bouchard
and Pham [2] extended duality techniques to characterize the optimal
solution. Blanchet-Scalliet et al. [1] considered a random horizon not
necessarily a stopping time. Zitković [21] looked at the problem of utility
maximization with a stochastic clock and an unbounded random endowment. Karatzas and Wang [14] treated utility maximization problem of
mixed optimal stopping/control type.
All these works that we mentioned above, assume a time horizon (deterministic or stochastic) and an evaluation criterion (expected utility)
which is optimized at time zero only. The possibility that the agent may
update the evaluation criterion as time goes by is not taken into account.
This may lead to time inconsistencies; it means that optimal strategies
decided at time zero may fail to remain optimal if the evaluation criterion is updated. On the other hand, the time consistent strategies will
be implemented in the future even though at later times the investor is
free to reconsider them.
Dynamic inconsistent behavior was first formalized analytically by
Strotz [20]. Recently the notion of forward utility appeared in the literature to deal with the issue of time consistency. Among other works, we
recall Choulli et al. [4], Musiela and Zariphopoulou [19] and Zitković
[22].
Choulli et al. [4] addressed the time consistency of portfolio problem in a semmimartingale setup where the investor preference is time
and state dependent. They required that the same trading strategy is
optimal for all stopping time investment horizons. This can be accomplished by considering a special utility random field. Choulli et al. [4]
provide such a utility random field, and it is generated by time and state
independent CRRA preferences.
This paper aims to find time and state dependent preferences that
fulfill the time consistency requirement of Choulli et al. [4]. In a Brownian motion framework, we search for a class of utility random fields
that are generated by a time and state independent utility which we call
seed utility. Furthermore, these utility random fields should satisfy the
following property: the same trading strategy is optimal (with respect to
TIME CONSISTENT UTILITY MAXIMIZATION
723
the expected utility criterion) for all stopping time investment horizons.
The class of seed utilities that we consider contains CRRA utilities as a
special case, and it can be more general.
It is interesting to point out that the time consistent random fields
that we construct are related to the coefficient of relative risk aversion
and coefficient of prudence of the seed utility. The coefficient of prudence
was introduced by Kimball [15] as a measure of the sensitivity to the
choices of risk. In proving our results we proceed in two steps. In the
first step, we solve the problem of optimal investment with a stopping
time horizon by using the martingale method. In the second step, we
require that for all stopping times horizons the same investment strategy
is optimal and this leads to the desired utility random field.
One direct application of our findings to the classical utility theory
is a novel portfolio decomposition formula. We show that the optimal
investment strategy of a time and state independent utility, although
not time consistent, can be decomposed as the sum of a horizon independent (myopic) component and a horizon dependent component. For
the special case of logarithmic utility, the horizon dependent component
is zero, which shows that the optimal portfolio for logarithmic utility
is myopic. An open problem is to find other classes of time consistent
utilities random fields besides those generated by seed utilities.
The remainder of this paper is structured as follows. In Section 2,
we introduce the financial market model and Section 3 presents the
objective. Sections 4 contains the main result of the paper, which is
proved in Section 5. Section 6 provides the portfolio decomposition
formula. We conclude with Section 7.
2 Model description
2.1 Financial market We adopt a model for the financial market consisting of one riskless bond and d stocks. We work in discounted terms,
that is the price of bond is constant and the stock price per share satisfies
the following stochastic differential equation (SDE)
n
X
dSi (t) = Si (t) αi (t) dt +
σij (t) dWj (t) ,
0 ≤ t ≤ ∞,
i = 1, . . . , d.
j=1
Here, W = (W1 , · · · , Wn )T is an n-dimensional Brownian motion on
a filtered probability space (Ω, {Ft }0≤t≤∞ , F, P), where {Ft }0≤t≤∞ is
the completed filtration generated by W. We assume that d ≤ n, i.e.,
there are at least as many sources of uncertainty as assets. As usual,
724
T. A. PIRVU AND U. G. HAUSSMANN
{α(t)}t∈[0,∞) = {(αi (t))i=1,··· ,m }t∈[0,∞) is an Rm valued mean rate of
return process, and
j=1,··· ,n
{σ(t)}t∈[0,∞) = {(σij (t))i=1,···
,m }t∈[0,∞)
is an m×n-matrix valued volatility process. These processes are assumed
to be progressively measurable with respect to filtration {Ft }0≤t≤∞ .
Assumption 2.1 The process α is uniformly bounded and the volatility
matrix σ has full rank. Moreover, σσ T is assumed uniformly elliptic, i.e.,
KId ≥ σσ T ≥ Id .
This assumption implies the existence of the inverse (σ(t)σ T (t))−1 ,
and that the market price of risk process
(2.1)
θ̃(t) = σ T (t)(σ(t)σ T (t))−1 α(t),
is uniformly bounded. At this point let us introduce an exogenous deterministic horizon T. The stochastic exponential process
(2.2)
e , exp
Z(t)
−
Z
t
1
θ̃ (u) dW (u) −
2
T
0
Z
t
2
k θ̃(u) k du
0
is a (true) martingale. Thus by the Girsanov theorem (Section 3.5 in
[13])
(2.3)
f (t) = W (t) +
W
Z
t
θ̃(u) du
0
is a Brownian motion under the equivalent martingale measure
(2.4)
e
e )1A ],
Q(A)
, E[Z(T
A ∈ FT .
Definition 2.1. We denote by M the set of probability measures which
satisfy:
2
(i) Q P and dQ
dP ∈ LT (P);
(ii) S is a local martingale under Q on [0, T ].
e ∈ M. The set M is
In the light of boundedness of θ̃, it follows that Q
usually referred to as the set of martingale measures. Standard results
show that a market model is arbitrage free as long as the set of martingale measures is nonempty. Therefore, our market model is arbitrage
TIME CONSISTENT UTILITY MAXIMIZATION
725
free. Let us point out that our model is also incomplete. Indeed, by
d ≤ n, the market price of risk system of equations
σ(t)θ(t) = α(t)
has multiple solutions θ(t) which can lead to other martingale measures
besides the one defined in (2.4). These martingale measures are defined
as in (2.4) via (2.2). The second fundamental theorem of asset pricing
asserts that a market model is complete if and only if the martingale
measure is unique. Consequently, our market model is incomplete.
2.2 Portfolio and wealth processes A (self-financing) portfolio is
defined as a pair (x, π). The constant x, exogenously given, is the initial
value of the portfolio, and π = (π1 , . . . , πd ) is a progressively measurable
process which satisfies the following integrability condition
(2.5)
P
X
d X
n Z
i=1 j=1
T
0
2
πi2 (t)Si2 (t)σij
(t) dt
< ∞ = 1.
Moreover, π(t) = (π1 (t), . . . , πd (t)) specifies how many units of the stock
i are held in the portfolio at time t. The leftover wealth at time t is
invested in the riskless bond. Since we work in discounted terms (the
bond price is used as a numeraire), the wealth process of the portfolio
(x, π) at time t is given by
Z t
(2.6)
X x,π (t) = x +
π(u) dS(u).
0
2.3 Risk preferences Having introduced the financial market let us
now turn to investor risk preference. As pointed out in the introduction
we consider utility random fields. Following [22] we have the following
formal definition.
Definition 2.2. A mapping U : [0, ∞) × R × Ω → R is called a utility
random field if it satisfies:
•
•
utility conditions: the mapping x → U(t, x, ω) is strictly concave,
strictly increasing function;
path regularity: the process t → U(t, x, ω) is progressively measurable.
For the sake of simplicity we write U(t, x) suppressing ω from the
notations.
726
T. A. PIRVU AND U. G. HAUSSMANN
3 Objective We want to find utility random fields U which satisfy
the requirement that the same trading strategy is optimal (with respect
to the expected utility criterion) for all stopping times investemnt horizons. Thus, the goal is to find a utility random field U and a portfolio
(x, π̂) such that
(3.1)
sup E U(τ, X x,π (τ )) = E U(τ, X x,π̂ (τ ))
π
for every stopping time τ adapted to the filtration {Ft }0≤t≤∞ with τ ≤
T. Here, the supremum is taken over the set of admissible portfolios,
which is to be defined, and π̂ is an admissible portfolio.
Inspired by Choulli et al. [4], we search for a special choice of U to
satisfy (3.1). More precisely, given an exogenous time and state independent utility U called “seed utility,” consider the class of utility random
fields of the from
(3.2)
U(t, x, ω) = U (x + V (t, ω)),
where the process V (t, ω) is of finite variation, progressively measurable
with respect to {Ft }0≤t≤T , and it satisfies V (0, ω) = 0. The utility
random field U inherits monotonocity and concavity form U. At first
glance, the choice of U in (3.2) may look artificial. If V ≡ 0, then U
is a time and state independent utility function. This type of utilities
can not satisfy the requirement of (3.1). The next choice, in our view,
is perhaps to look for U generated by a “seed utility” U and a process
V, as in (3.2). The process V can be seen as a time and state correction
applied to the “seed utility” U via (3.2).
Let us now turn to the seed utility U. It is assumed that U : (0, ∞) →
R is strictly increasing and strictly concave. Moreover, we restrict ourselves to utility functions U which are 4-times continuous differentiable
and satisfy the Inada conditions
(3.3)
U 0 (0+) , lim U 0 (x) = ∞,
x↓0
U 0 (∞) , lim U 0 (x) = 0.
x↑∞
We shall denote by I(·) the (continuous, strictly decreasing) inverse of
the marginal utility function U 0 (·). By (3.3)
(3.4)
I(0+) , lim I(x) = ∞,
x↓0
I(∞) , lim I(x) = 0.
x↑∞
TIME CONSISTENT UTILITY MAXIMIZATION
727
Let us introduce the Legendre transform of −U (−x)
(3.5)
e (y) , sup[U (x) − xy] = U (I(y)) − yI(y),
U
0 < y < ∞.
x>0
e is strictly decreasing, strictly convex and satisfies the
The function U(·)
dual relationships
e 0 (y) = −x
U
(3.6)
and
(3.7)
iff U 0 (x) = −y,
e (y) + xy] = U(U
e 0 (x)) + xU 0 (x),
U (x) = inf [U
y>0
0 < x < ∞.
Assumption 2.2
(3.8) y 2 |I 00 (y)| ∨ (−yI 0 (y)) ∨ I(y) < k1 y −β + k2
for every y ∈ (0, ∞),
for some positive constants k1 , k2 and β. Here, a ∨ b = max(a, b).
The process V (t, ω), which for simplicity we denote V (t), is required
to satisfy the following integrability condition
(3.9)
E
sup |V (t)|
0≤t≤T
2
< ∞.
At this point we need to be more specific about the admissible trading
portfolios. Given the time horizon τ, the class of admissible portfolios is
(3.10)
AV (τ ) , {π | X x,π (t) + V (t) > 0, P a.s., 0 ≤ t ≤ τ
and E[U (X x,π (τ ) + V (τ ))]− < ∞}.
The admissible trading portfolios satisfy a non bankruptcy type condition up to the correction term V.
We are now ready to reformulate the problem (3.1) taking into account the special structure of U given in (3.2). Our objective is to find
a finite variation process V and an admissible portfolio (x, π̂) such that
(3.11)
sup EU (X x,π (τ ) + V (τ )) = EU (X x,π̂ (τ ) + V (τ ))
π∈AV (τ )
for every stopping time τ with τ ≤ T.
728
T. A. PIRVU AND U. G. HAUSSMANN
4 The main result The following Theorem is our main result.
Before stating it, we need to introduce the following function
1 00
I (z)z 2 + I 0 (z)z.
2
Theorem 4.1. Let U be a utility function which satisfies (3.3), (3.8);
let x be the initial wealth, and let the process V be defined by
Z t
e
(4.2)
V (t) ,
F (U 0 (x)Z(u))k
θ̃(u)k2 du, 0 ≤ t ≤ T.
(4.1)
F (z) =
0
Then, the portfolio (x, π̂), given by
(4.3) (π̂(t))i , −
1
e
e
((σ(t)σ T (t))−1 I 0 (U 0 (x)Z(t))
Z(t)α(t))
i,
Si (t)
0 ≤ t ≤ T, i = 1, · · · , d,
is admissible, i.e., π̂ ∈ AV (τ ) for every stopping time τ ≤ T. Moreover,
it follows that
(4.4)
sup EU (X x,π (τ ) + V (τ )) = EU (X x,π̂ (τ ) + V (τ )),
π∈AV (τ )
for every stopping time τ with τ ≤ T.
Remark 4.2. If the seed utility is logarithmic, i.e., U (x) = log x, then
V ≡ 0. Being optimal for the logarithmic utility, the vector π̂ satisfies
(4.5)
(π̂(t))i =
(ζM (t))i X π̂ (t)
,
Si (t)
i = 1, . . . , d,
with ζM (t) , (σ(t)σ T (t))−1 α(t) being the Merton proportion. The future evolution of S does not enter in the formula (4.3) and (4.5), hence
we refer to π̂ as a myopic portfolio.
Direct computations show that F defined in (4.1) is in fact
y2
U 00 (I(y))
U 000 (I(y))
F (y) = 00
+
2
−
,
[U (I(y))]2
U 00 (I(y))
U 0 (I(y))
and it has the following economic interpretation: F (y) is the difference
000
(I(y))
between the coefficient of prudence, − UU 00 (I(y))
, and twice the coefficient
00
(I(y))
. The coefficient of prudence reflects
of relative risk aversion, − UU 0 (I(y))
an individual’s propensity to take precautions when faced with risk. The
coefficient of relative risk aversion goes back to Arrow-Pratt and reflects
the tendency to avoid risk altogether.
TIME CONSISTENT UTILITY MAXIMIZATION
729
5 Proof of the main result Before we proceed with the proof of
this theorem, we need the following preparatory lemmas.
Lemma 5.1. The wealth process X x,π (see equation (2.6)) associated
with an admissible portfolio π ∈ AV (τ ) is a Q-supermartingale for every
Q ∈ M.
Proof. Let Tn ↑ T be a localizing sequence, u ≤ t ≤ τ, and let EQ
be the expectation operator with respect to a given martingale measure
Q. Hölder’s inequality and (3.9) imply that EQ [sup0≤t≤T |V (t)|] < ∞.
Then, Fatou’s lemma yields
EQ [X x,π (t) + sup |V (s)| |Fu ]
0≤s≤t
≤ lim inf EQ [X x,π (t ∧ Tn ) + sup |V (s)| |Fu ]
n→∞
0≤s≤t
= X x,π (u) + EQ [ sup |V (s)| |Fu ],
0≤s≤t
whence
EQ [X x,π (t) |Fu ] ≤ X x,π (u).
(5.1)
Therefore, X x,π is Q-supermartingale.
Lemma 5.2. For every number β
e β < ∞.
(5.2)
E sup Z(t)
0≤t≤T
Proof. If β is positive, Burkholder-Davis-Gundy (see [13, p. 166]) in
conjunction with boundedness of θ̃ and Hölder’s inequality yield the
claim. If β is negative
Z t
β
−β
2
e
Z(t) = M (t) exp
−β k θ̃(u) k du ,
0
where M (t) is the local martingale
M (t) , exp
Z
t
θ̃T (u) dW (u) −
0
and the same argument applies.
1
2
Z
t
0
k θ̃(u) k2 du ,
730
T. A. PIRVU AND U. G. HAUSSMANN
Lemma 5.3. Assume that there exists a portfolio process π̂ such that
π̂ ∈ AV (τ ) for every stopping time τ ≤ T, and
e
X x,π̂ (t) + V (t) = I(U 0 (x)Z(t)),
(5.3)
Then
(5.4)
sup
0 ≤ t ≤ T.
EU (X x,π (τ ) + V (τ )) = EU (X x,π̂ (τ ) + V (τ ))
π∈AV (τ )
for every stopping time τ ≤ T.
Proof. Let π be a portfolio such that π ∈ AV (τ ) for every stopping
time τ ≤ T. Then, by Lemma 5.1, the associated wealth process X x,π ,
e supermartingale. Problem 3.26, p. 20 in [13] implies that
is a Q
e )X x,π (τ )] ≤ x.
E[Z(τ
(5.5)
Recall that
e
X x,π̂ (t) = I(U 0 (x)Z(t))
− V (t),
cf. (5.3). The process X x,π̂ , being a stochastic integral with integrator
e
S, is a local Q-martingale.
In the light of Lemma 5.2 and (3.8), one
obtains
(5.6)
E
2
e
sup |I(U (x)Z(t))|
0
0≤t≤T
< ∞.
Consequently, by (3.9) and (5.3), it follows that
E
sup |X
0≤t≤T
x,π̂
(t)|
2
< ∞.
e sup0≤t≤T |X x,π̂ (t)| < ∞, where E
e
Hölder’s inequality implies that E
denotes the expectation operator with respect to the martingale measure
e Therefore, by the Dominated Convergence Theorem, the process X x,π̂
Q.
e
is a (true) Q-martingale.
Hence, for any stopping time τ ,
(5.7)
e )X x,π̂ (τ )] = x,
E[Z(τ
TIME CONSISTENT UTILITY MAXIMIZATION
731
by [13, Problem 3.26, p. 20]. Due to the concavity of the seed utility
function U and (5.3), it follows that
U (X x,π (τ ) + V (τ )) − U (X x,π̂ (τ ) + V (τ ))
≤ U 0 (X x,π̂ (τ ) + V (τ ))(X x,π (τ ) − X x,π̂ (τ ))
e )(X x,π (τ ) − X x,π̂ (τ )).
= U 0 (x)Z(τ
Taking expectation and using (5.5) and (5.7), we complete the proof.
Proof of Theorem 4.1. First we claim that
(5.8)
e
X x,π̂ (t) + V (t) = I(U 0 (x)Z(t)),
0≤t≤T
for the portfolio π̂ given by (4.3) and the process V given by (4.2).
Indeed, by Itô Lemma and (4.2)
(5.9)
e
I(U 0 (x)Z(t))
=x−
Z
t
0
+
=x−
Z
Z
t
0
e
e θ̃(u) dW
f (u)
U 0 (x)I 0 (U 0 (x)Z(u))
Z(u)
t
0
e
F (U 0 (x)Z(u))k
θ̃(u)k2 du
e
e θ̃(u) dW
f (u) + V (t).
U 0 (x)I 0 (U 0 (x)Z(u))
Z(u)
For π̂ given by (4.3), it follows that
(5.10)
X x,π̂ (t) = x −
Z
t
0
e
e θ̃(u) dW
f (u).
U 0 (x)I 0 (U 0 (x)Z(u))
Z(u)
Equations (5.9) and (5.10) imply (5.8). Next, let us prove that the
portfolio (x, π̂), given by (4.3) is admissible, i.e., π̂ ∈ AV (τ ) for every
stopping time τ ≤ T. It follows from (3.5) that
U (I(y)) ≥ U (x) + y[I(y) − x],
∀x > 0, y > 0.
This inequality in conjunction with (3.8) gives
(5.11)
U − (I(y)) < k2 + k3 y 1−κ ,
732
T. A. PIRVU AND U. G. HAUSSMANN
for some positive k2 and k3 . Inequality (5.11) and the boundedness of θ̃
imply that for every τ stopping time
e ))]− < ∞.
E [U (X x,π̂ (τ ) + V (τ ))]− = EU [I(U 0 (x)Z(τ
Moreover, by (3.4) and (5.8), it follows that
X x,π̂ (t) + V (t) > 0, P a.s.,
0 ≤ t ≤ T.
Hence, π̂ ∈ AV (τ ) for every τ stopping time. Then, by (5.8) and
Lemma 5.3, it follows that
sup
E U (X x,π (τ ) + V (τ, x)) = EU (X x,π̂ (τ ) + V (τ, x)),
π∈AV (τ,x)
for every stopping time τ ≤ T,
which concludes the proof.
Next, we compute the process V for a special choice of seed utility.
Let U (x) = xp /p, with p < 1 the coefficient of relative risk aversion.
Then, by (4.2),
(5.12)
V (t) =
xp
2(p − 1)2
Z
t
0
1
e
p−1 kθ̃(u)k2 du.
[Z(u)]
This shows that the process V can be either positive or negative. A
generalization of (5.12) was obtained by Choulli et al. [4] in a semimartingale framework.
6 A portfolio decomposition formula Let us work now in the
context of complete markets, i.e., n = d, and the objective is to maximize a time and state independent utility U of the final wealth at some
exogenous random time horizon τ. In order to compute the optimal investment strategy, one can apply Theorem 4.1. We can regard V (τ )
as the payoff of a contingent claim that we can hedge using the stocks.
Indeed, the martingale process
e [V (τ )|Ft ]
V (t) , E
admits the stochastic integral representation
Z t
e (τ ) +
f (u)
(6.1)
V (t) = EV
β τ (u) dW
0
0≤t≤τ
TIME CONSISTENT UTILITY MAXIMIZATION
733
Rτ
for some Ft -adapted process β which satisfies 0 kβ(u)k2 du < ∞ a.s.
(e.g., [13, Lemma 1.6.7]). Furthermore, we want to find the vector
valued process π̄ such that
Z t
Z t
T
f
(6.2)
β (s) dW (s) =
π̄ T (s) dS(s),
0 ≤ t ≤ τ.
0
0
This can be accomplished as follows. Let A be the n × n matrix with the
entries Aij = Si σij . Since the volatility matrix σ has linearly independent rows, the matrix A has linearly independent rows, so rank A = n.
Therefore,
π̄ = (AT )−1 β,
(6.3)
satisfies (6.2). In the present context, some part of the initial wealth
amount x is needed to finance the contingent claim V (τ ). Therefore, we
apply Theorem 4.1 with the modified initial wealth x∗ defined by
Z t
0
2
e
e
(6.4) x∗ , inf z ≥ 0 | z + E
F (U (z)Z(u))kθ̃(u)k du = x .
0
Theorem 4.1 provides the following portfolio decomposition formula:
given the initial wealth x, the agent should hold π̂ + π̄ shares of stocks,
where
(6.5) (π̂(t))i = −
1
e
e θ̃(t)T )i ,
((σ T (t))−1 U 0 (x∗ )I 0 (U 0 (x∗ )Z(t))
Z(t)
Si (t)
0 ≤ t ≤ τ, i = 1, · · · , d,
is the myopic portfolio, π̄ of (6.3) is the hedging portfolio, and x∗ is given
by (6.4). This investment strategy is optimal, i.e.,
(6.6)
sup EU (X x,π (τ )) = EU (X x,π̂+π̄ (τ )).
π
Remark 6.1. The myopic portfolio does not depend on the future evolution of the stock prices. Moreover it has the same functional form of
(6.5) for all stopping times horizons τ. This depends on the random horizon τ only through x∗ of (6.4). For this reason the myopic portfolio can
be easily computed. The hedging portfolio can be obtained by a stochastic representation formula via (6.2) and (6.3). Some complications may
arise in computing the hedging portfolio because stopping times are not
Malliavin differentiable. In some situations, e.g., U (x) = x /, with
small, the myopic portfolio can be an approximation of the optimal
portfolio.
734
T. A. PIRVU AND U. G. HAUSSMANN
7 Conclusion In this paper we provide utility random fields that
lead to a time consistent expected utility criterion. We work in an incomplete market, and the randomness is driven by a multidimensional
Brownian motion. We search for utility random fields generated by a
time/state independent utility and a process of finite variation. The
utility an agent gets from wealth is time/state dependent and equals to
a time/state independent utility applied to the wealth corrected by a
finite variation process. This process inherits properties from the financial market and the time/state independent utility function. A portfolio
decomposition formula that splits the optimal investment into the myopic portfolio and hedging portfolio is provided. Many avenues are still
open for future research. We provide time consistent utility random
fields that have a special structure, so it appears natural to try to find
other classes of time consistent utility random fields as well. Numerical
experiments showing the behavior of the myopic and hedging portfolios
in some examples can bring insight into the optimal portfolios.
REFERENCES
1. C. Blanchet-Scalliet, N. El Karoui, M. Jeanblanc and L. Martellini, Optimal investment and consumption decisions when time horizon is uncertain, J. Econ.
Dyn. Control 29 (2005), 1737–1764.
2. B. Bouchard and H. Pham, Wealth-path dependent utility maximization in
incomplete market, Finance Stoch. 8 (2004), 579–603.
3. T. Choulli and C. Stricker, Minimal entropy-Hellinger martingale measures in
incomplete markets, Math. Finance 15 (2005), 465–490.
4. T. Choulli, C. Stricker and J. Li, Minimal Hellinger martingale measures of
order q, Finance Stoch. 11 (2007), 399–427.
5. J. C. Cox and C. H. Huang, Optimal consumption and portfolio policies when
asset prices follow a diffusion process, J. Econ. Theory 49 (1989), 33–83.
6. I. Ekeland and A. Lazrak, Being serious about non-commitment: subgame
perfect equilibrium in continuous time, preprint, (2006).
7. I. Ekeland and T. A. Pirvu, Investment and consumption without commitment,
Math. Finance Econ. 2 (2008), 57–86.
8. I. Ekeland and T. A. Pirvu, Portfolio management with non-exponential discounting, (2009), submitted.
9. I. Ekeland, O. Mbodji and T. A. Pirvu, On the pensioner’s management problem: A time consistent approach, (2010), working paper.
10. T. Goll and J. Kallsen, A complete explicit solution to the log-optimal portfolio
problem, Ann. Appl. Probab 13 (2003), 774–799.
11. H. He and D. H. Pearson, Consumption and portfolio policies with incomplete
markets and short-sale constraints: The infinite dimensional case*, J. Econ.
Theory 54 (1990), 259–304.
TIME CONSISTENT UTILITY MAXIMIZATION
735
12. I. Karatzas, J. P. Lehoczky and S. E. Shreve, Optimal portfolio and consumption decisions for a “small investor” on a finite horizon, SIAM J. Control
Optim. 25 (1987), 1557–1586.
13. I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, 2nd
ed., Springer-Verlag, New York, 1991.
14. I. Karatzas and H. Wang, Utility maximization with discreptionary stopping,
SIAM J. Control Optim. 39 (2000), 306–329.
15. M. Kimball, Precautionary saving in the small and in the large, Econometrica
58 (1990), 53–73.
16. D. Kramkov and W. Schachermayer, The asymptotic elasticity of utility functions and optimal investment in incomplete markets, Ann. Appl. Probab. 9
(1999), 904–950.
17. D. Levhari and J. Mirman, Savings and consumption with an uncertain horizon, J. Polit. Econ. 85 (1977), 265–281.
18. R. C. Merton, Optimum consumption and portfolio rules in a conitinuous-time
model, J. Econ. Theory 3 (1971), 373–413.
19. M. Musiela and Zariphopoulou, Investment performance measurement, risk
tolerance and optimal portfolio choice, (2009), to appear.
20. R. Strotz, Myopia and inconsistency in Dynamic Utility Maximization, Rev.
Finan. Stud. 23 (1955), 165–180.
21. G. Zitković, Utility Maximization with a stochastic clock and an unbounded
random endowment, Ann. Appl. Probab. 15 (2005), 748–777.
22. G. Zitković, A dual characterization of self-generation and exponential forward
performances, Ann. Appl. Probab. 19 (2009), 2176–2210.
Dept of Mathematics and Statistics, McMaster University
1280 Main Street West, Hamilton, ON, L8S 4K1.
E-mail address: [email protected]
Dept of Mathematics, University of British Columbia,
Vancouver, BC, V6T 1Z2.
E-mail address: [email protected]