Research Collection
Doctoral Thesis
The optimal martingale measure for investors with exponential
utility function
Author(s):
Steiger, Gallus Johannes
Publication Date:
2005
Permanent Link:
https://doi.org/10.3929/ethz-a-005047932
Rights / License:
In Copyright - Non-Commercial Use Permitted
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ETH Library
Diss. ETH No. 16006
The
Optimal Martingale Measure for
Investors with Exponential Utility Function
A dissertation submitted to the
SWISS FEDERAL INSTITUTE OF TECHNOLOGY
ZURICH
for the
of
degree
Doctor of Mathematics
presented by
GALLUS JOHANNES STEIGER
dipl.
born
citizen of
accepted
math.
September 15,
Meggen &
Schlierbach
Prof. Dr. F.
Prof. Dr. J.
Delbaen,
Kallsen,
examiner
co-examiner
Rheinländer,
Prof. Dr. M.
(LU)
the recommendation of
on
Prof. Dr. T.
1972
Schweizer,
2005
co-examiner
co-examiner
Für
Saskia,
Fiona und Meret
Acknowledgments
I would like to express my
gratitude to Thorsten Rheinländer for his support, patience,
throughout my doctoral studies. It is not often that one finds an
advisor and colleague who always finds the time for listening to the little problems and
roadblocks that unavoidably crop up in the course of doing research. His technical and
editorial advice was essential to the completion of this dissertation.
and encouragement
thanks also go to
My
on
my dissertation.
In
Freddy Delbaen. He made it possible for me to work full-time
addition, he readily accepted to supervise me after Thorsten left
Zürich to teach at the London School of Economics. I
the examination committee, Jan Kallsen and Martin
previous drafts. Their feedback
substantially.
I
am
mathematics and
hospitality
while
improved
also thankful to
helped
visiting
me
find my
owe
him and the other members of
Schweizer,
many thanks for
reading
the presentation and contents of this dissertation
Rüdiger Frey who led me into modern financial
subject. I want to thank Friedrich Hubalek for his
Vienna.
My gratitude goes to the people of the Financial Services Risk Management group of
Ernst & Young, who let me work part-time on my dissertation. Their flexibility and
enduring support allowed me to do research. I am also grateful to the assistance-group 3
of the Mathematics Departement at the ETH Zürich. It was always a pleasure being part
of this group.
The
biggest thank, however, goes to my family. To Saskia, my
encouraged and supported me writing this dissertation and to
precious daughters.
Financial support
is
by NCCR
gratefully acknowledged.
Financial Valuation and Risk
v
wife and best
Fiona and
Management
friend, who
Meret, my two
and Credit Suisse
VI
Abstract
The
problem of valuing and hedging financial products in incomplete markets, in which
non-replicable financial positions containing some intrinsic risk, is one of the
main problems in financial mathematics. The approach, which has been chosen in this
thesis, starts from a rational investor, who aims to maximize his expected exponential
there exist
utility. Via utility indifference arguments, financial instruments can be valued and the
corresponding hedging strategy can uniquely be defined. It is well known that, in a semimartingale model, the solution can be determined via a stochastic optimization problem
where we have to maximize a concave functional on some space of stochastic integrals. Al¬
ternatively we may consider the dual problem where we have to minimize the conjugated
Whereas representa¬
convex functional on the space of equivalent martingale measures.
tion properties of the optimal martingale measures are known, explicit solutions are at
hand only for some special cases.
In this
we
present
a
method how to solve the stochastic optimization
problem
in
Starting from the known representation properties, we
"optimal martingale measure equation". This equation helps to
guess the optimal martingale measure which then has to be verified. Several special cases
are treated. We discuss the valuation and hedging problem in case of a financial derivative
on an underlying whose return is modeled as an additive process. We show that the case
of an illiquid underlying, where hedging can only be performed by correlated assets, may
be treated in almost the same way. A related problem is the identification of the so-called
"minimal entropy martingale measure" which we determine for a wide class of stochastic
volatility models.
case
of
thesis,
jump-diffusion
determine
a
processes.
so-called
We show that for the models mentioned
above, the optimal martingale measure will
be identified by a so-called "interactive partial differential equation". Using FeynmanKac results and the Picard-iteration method, we establish existence and uniqueness of a
classical solution.
vn
Vlll
Kurzfassung
Bewertung und Absicherung von Finanzprodukten im Falle von un¬
vollständigen Märkten, in welchen zufallsbehaftete Finanzpositionen mit unvermeidbaren
intrinsischen Risiken auftreten können, ist ein zentrales Problem in der Finanzmathe¬
Das Problem der
matik.
Der in dieser Arbeit
dessen Ziel die
von
gewählte Ansatz geht von einem rationalen Investor aus,
Maximierung seines erwarteten exponentiellen Nutzens ist. Ausgehend
seinen Präferenzen können Finanzinstrumente nutzen-indifferent bewertet sowie die
entsprechende Absicherungsstrategie bestimmt
Lösung in einem Semimartingalmodell durch
dargestellt werden kann, wobei wir ein konkaves
stischen
maximieren haben.
werden.
Es ist
wohlbekannt,
ein stochastisches
dass diese
Optimierungsproblem
Funktional über einem Raum
von
stocha-
Alternativ als duales Problem
Integralen
ausgedrückt
konjugierte konvexe Funktional über dem Raum der äquivalenten Mar¬
tingalmasse minimieren. Während Darstellungseigenschaften des optimalen Martingalmasses bekannt sind, bestehen explizite Lösungen jedoch nur in einigen Spezialfällen.
zu
müssen wir das
vorliegenden Arbeit wird ein Verfahren aufgezeigt, wie sich das stochastische Op¬
timierungsproblem im Falle von Sprung-Diffusionsprozessen lösen lässt. Ausgehend von
den bekannten Darstellungseigenschaften wird eine "Optimal Martingale Measure Equa¬
tion" ermittelt. Anhand dieser Gleichung kann das optimale Martingalmass sozusagen
erraten werden, welches anschliessend verifiziert werden muss. Mehrere Spezialfälle wer¬
den behandelt. Wir diskutieren das Bewertungsproblem im Falle eines beliebigen Fi¬
nanzderivates auf einem Basisprodukt, dessen Rendite wir als additiven Prozess model¬
lieren. Wir zeigen zudem auf, dass der Spezialfall, in welchem das Underlying illiquid
ist und deshalb das Finanzderivat durch ein verwandtes Finanzinstrument abgesichert
werden muss, sehr ähnlich zu behandeln ist. Ein ähnliches Problem ist die Identifizierung
des sogenannten "minimalen Entropie-Masses", welches wir für eine grosse Klasse von
In der
stochastischen Volatilitätsmodellen ermitteln.
zeigen auf, dass für die obengenannten Modelle das optimale Martingalmass anhand
einer sogenannten "interaktiven partiellen Differentialgleichung" identifiziert wird. Mit¬
Wir
tels
und
Feynman-Kac-Resultaten und der Iterationstechnik
Eindeutigkeit einer klassischen Lösung.
ix
von
Picard
zeigen
wir Existenz
X
Contents
1
2
3
4
5
6
Introduction
1
1.1
Thesis
1.2
Overview of Results
1.3
Contribution and
Subject
1
General Results for
2
Organization
Exponential Utility
2.1
General
2.2
Utility
2.3
Zero
2.4
Verification Procedure
Indifference
and
Pricing
Martingale Properties
5
9
Framework
Semimartingale
Marginal Utility
9
Hedging
13
Price
15
16
of
Jump-Diffusion
3.1
Filtration and Asset Model
3.2
Class of
3.3
Optimal Martingale
3.4
Example:
Equivalent Martingale
Measure
Models
Measures
Equation
The MEMM in the Additive Process Case
4.1
Solutions for Parabolic
4.2
Solution for First-Order
19
19
Interactive Partial Differential
Martingale
of this Thesis
Equations
Equations
5.2
Motivation and Main Theorem
5.3
Discussion of Function
5.4
Existence of Classical Solution
5.5
Optimal Martingale
5.6
Examples
25
31
31
Measures under Additive Processes
Preliminaries
23
43
Equations
5.1
21
51
51
53
WL
58
66
Measure Verification
of k,\ and K2
MEMM for Stochastic
75
77
Volatility
Models
Assumptions
83
6.1
Model and
6.2
Main Theorem
85
6.3
Proof of the Main Theorem
88
6.4
The
6.5
The Barndorff-Nielsen
6.6
Existence of
Orthogonal Volatility
u
84
Process
Shephard
92
Model
in the BN-S model
94
98
XI
CONTENTS
Xll
Appendix
107
A
107
Levy
Processes
A.l
The
A.2
The
Lévy-Khintchine Representation
Lévy-Itô Decomposition
107
110
Index of Notation
113
Bibliography
117
Curriculum Vitae
123
Chapter
1
Introduction
1.1
Thesis
In this thesis
hedging
and
we
Subject
consider
optimal
a
risk-averse market participant who is confronted with
investment
problems
in
incomplete
pricing,
markets.
If the
payoff of a contingent claim is replicable by dynamic trading, the claim can be per¬
fectly hedged by the replicating strategy, and its price is defined as the replication cost.
Since being dynamically redundant, such claims have little benefit from a financial point
of view. If a contingent claim is not redundant, it incorporates some unavoidable risk.
Preferences have to be introduced in order to evaluate optimal prices and hedging strate¬
gies. We adapt the perspective of a rational market participant who aims to maximize
his expected utility according to his level of risk aversion. Based on these preferences, the
utility indifference price is that price where the decision maker is indifferent in terms of
maximal expected utility between holding the contingent claim or not. In this sense, the
utility indifference pricing problem is related to the solution of the utility maximization
problem with an additional liability. The hedging strategy for the contingent claim is
defined as the adjustment of the optimal portfolio strategy, induced by the additional
liability from the claim. Mathematically speaking, we are confronted with a stochastic
optimization problem in which concave functionals are maximized on spaces of stochastic
integrals. We look for both the maximal value but also the corresponding optimal strategy.
-
-
preferences of the market participant may be described by the expo¬
nential utility function, motivated by its suitable features. The theoretical basis for the
exponential utility function is a condition called constant risk aversion. This condition
holds when it is true that whenever all possible outcomes of any uncertain alternative
are changed by the same specified amount, the decision maker's certainty equivalent for
We
assume
that the
the alternative also
that
of the
exponential utility
function,
problem
utility
Legendre transform, be solved
by considering its dual problem, namely to find a martingale measure minimizing some
functional and whose density process can be written in a particular form. A constructive
characterization allows to uniquely describe the optimal martingale measure. Due to the
the
better
tractability,
problem.
changes by
same
amount.
a
common
case
can, via the
maximization
it is
In
to concentrate
strategy
1
on
the solution for the dual
CHAPTER 1.
2
INTRODUCTION
Many different forms of incompleteness are imaginable. To obtain more constructive and
explicit results, we restrict ourselves to jump-diffusion asset models. More specifically,
we consider asset processes which are adapted to a filtration which is generated by a
Levy process. For two well-studied model classes, stochastic optimization problems are
analyzed:
•
Pricing
and
hedging
a
contingent claim under additive
A market participant wants to sell
is modeled
assuming
•
by
an
processes:
European contingent claim on some asset which
additive log-return process. The problem is further generalized by
a
that the asset itself is not tradable.
Portfolio optimization in
An investor is
case
of stochastic
volatility
optimizing his portfolio consisting of
set, modeled with
processes:
a
riskless bond and
some
as¬
volatility process, and allowing for jumps in
volatility
By a duality argument, the stochas¬
tic optimization problem corresponds to identifying the so-called minimal entropy
martingale measure.
the
some
stochastic
and the asset process itself.
though both problems look fairly different, a
sketching this approach, we give a short overview
Even
similar
approach
of what had
may be used. Before
already
been achieved in
the literature.
Overview of Results
1.2
Since there
replicate perfectly a non-redundant contingent claim, a
market participant willing to hedge is exposed to some hedge mismatch and there is no
unique valuation. In the following, we want to present an overview of existing results, at
the same time supporting the approach chosen in this thesis.
Pricing
are
and
strategies
no
Hedging
in
to
Incomplete
Markets
charge a super-replication price for selling an
risks, as studied in El Karoui and Quenez
option
(1995), Kramkov (1996) and Föllmer and Kabanov (1998). In general, the price is usually
forbiddingly high. Eberlein and Jacod (1997) investigated this issue in pure-jump models,
and Bellamy and Jeanblanc (2000) studied it in jump-diffusion models. In both cases, the
superreplication price for a European option is the trivial upper bound of the no-arbitrage
interval. Take the most common example of a call option. The super-hedging strategy is
to buy and hold, and therefore the price of the call is equal to the initial stock price which
is too expensive. Since super-hedging is not a realistic solution, the trader is restricted
to charge a reasonable price, find a partial hedging strategy according to some optimality
Let
us
assume
so
that the trader decides to
that he
can
criterion, and bear
Several
is based
on
some
risks in the end.
identifying solutions of pricing and hedging
circumstances. Becherer (2001) gives a brief overview. A prominent approach
utility indifference pricing. Considerations of this sort go back to the 'principle
approaches
under such
trade to eliminate all
have been
developed
for
OVERVIEW OF RESULTS
1.2.
of
equi-marginal utility'
3
formulated
by
in the economics literature. The idea behind
willing
is
to pay
a
certain amount
today
for
(1871)
and have been
extensively developed
utility indifference pricing is that an investor
the right to receive the claim such that he is
Jevons
expected utility terms than he would have been without the claim. The
corresponding utility indifference hedging strategy is defined as the adjustment of the is¬
suer's optimal portfolio strategy that is induced by the additional liability from the claim.
no
worse
off in
Mathematically, the problem is closely related with the stochastic portfolio optimization
problem under an additional liability. Hodges and Neuberger (1989) were the first to
adapt the static certainty equivalence concept to a dynamic financial market. Similar
ideas
are
also important in actuarial mathematics.
the premium
principle
exponential form,
see
of
equivalent utility
Gerber
(1979)
There is
a
valuation method called
which has desirable properties if
utility
is of
for details.
subjective price, which, due to diversifi¬
cation effects, is not linear in volume, i.e. the subjective value of two risky claims is not
twice the value of one risky claim. However, the true utility indifference price can be
linearly approximated by the 'zero marginal utility price'. Based on Davis (1997), the
price is chosen in such a way that the agent is locally indifferent to small positions in
the claim. Goll and Rüschendorf (2001) showed that the minimal distance martingale
measure, where the distance is induced by the convex conjugate of the utility function,
is the appropriate pricing measure for Davis' price proposition. In this sense, his price
proposition is consistent with the no arbitrage condition. However, we note that choosing
a particular pricing measure determines a valuation for a claim but does not settle the
hedging problem.
By
its
nature, the utility indifference price is
Stochastic Portfolio
Merton
whose
Optimization
a
Problem
(1969, 1971) investigated the portfolio optimization problem
is to maximize his
Founded
for
a
rational investor
the axiomatization of
expected utility.
by von Neumann and Morgenstern (1944) and Herstein and Milnor (1959), this type of rational preferences constitutes the reference model
in the theory of portfolio selection. Merton derived the optimal portfolio based on Bell¬
man's dynamic programming approach.
objective
uncertainty
choice under
on
introduced
recently, a martingale approach to the problem in complete Itô-process markets was
by Pliska (1986), Karatzas et al. (1987) and Cox and Huang (1989). They re¬
lated the marginal utility from the terminal wealth of the optimal portfolio to the density
of the (unique) martingale measure, using powerful convex-duality techniques. Difficul¬
ties with this approach arise in incomplete markets. The main idea here is to use the
convex nature of the problem, to formulate and solve a dual variational problem over the
set of martingale measures, and then proceed as in the complete case. In discrete time
and on a finite probability space, the problem was studied by He and Pearson (1991a).
In a continuous-time set-up, it was discussed by Karatzas et al. (1991), He and Pear¬
in a general semimartingale framework
Kramkov and Schachermayer
son (1991b) and
(1999) and Schachermayer (2001). In contrast to Merton's original approach, the martin¬
gale method avoids the assumption of Markovian asset processes which is necessary for
the classical Bellman approach.
More
introduced
-
-
CHAPTER 1.
4
INTRODUCTION
the
martingale method, the problem of utility maximization under an additional
liability has been analyzed by Delbaen et al. (2002) for the exponential utility function.
By the very nature of the approach, the solution is neither constructive nor explicit.
However, the optimal martingale measure must minimize some functional containing the
In
relative entropy and the expectation of the contingent claim under this measure.
case of a portfolio optimization problem, i.e. without an additional contingent claim, the
optimal martingale measure is therefore the one with minimal relative entropy, the socalled minimal entropy martingale measure (MEMM). This measure equals the minimal
distance martingale measure in case of the exponential utility function, and hence, the
pricing measure for the 'zero marginal utility price'.
Using
Results
Explicit
Concrete results for the combined
and investment
hedging
problem
are
available
so
far
only in very specific models, whereas most of these models are based on a Brownian
filtration, where the asset prices are driven by Brownian motions and the incompleteness
comes
from the fact that there
Davis
(2000),
more
detail results in
processes'
•
Henderson
are more
(2002),
(see
Brownian motions than assets
Rheinländer
(2005),
Hobson
(2004)).
Let
of jump-diffusion processes. We consider the two
case
and 'stochastic
volatility
for instance
us
discuss in
cases
'additive
models':
Additive Processes:
log-return process of the asset is additive. A vast
case of Levy processes (e.g. Chan (1999), Kallsen
(2000), Miyahara (2001), Fujiwara and Miyahara (2003), Esche and Schweizer (2005))
Hence, the MEMM in case of additive processes is partly identified. On the other
hand, utility indifference pricing of a contingent claim, i.e. the utility maximization
problem under an additional liability, has not been discussed at all.
In this case,
we assume
that the
literature discusses the MEMM in
•
Stochastic
Volatility
Models:
special type of stochastic volatility models to summarize
existing results. The stochastic volatility model of Barndorff-Nielsen and Shephard
(2001), hereafter termed the BN-S model, is characterized by simultaneous jumps
in the asset process and in the volatility process. In its simplest form, the model is
Let
us
first introduce
defined
a
as
JO
-^
do\
whereas B is
{/i
=
—Xa^_dt
+
motion,
Z is
Brownian
a
Gaussian component and
with A
In
>
case
process
process.
+
+
at.dBt
+
(epx-l)dZxt
dZxt,
only positive
a
subordinator
increments
-
-
and ß,
Levy process with no
ß, À, p are real constants
a
0 and p < 0.
of p
-
ßal}dt
=
St-
=
0,
in this
In
a
we
case
slight
call it
a
simplified
BN-S model.
the Brownian motion B
abuse of notation,
we
only
-
is
call such
The risk driver of the asset
independent
a
model
a
of the
volatility
'continuous asset
CONTRIBUTION AND ORGANIZATION OF THIS THESIS
1.3.
process with
such
independent
Whereas,
volatility'.
Grandits and Rheinländer
model,
a
stochastic
(2002)
in
5
general
MEMM,
version of
a
determined the
Benth and
special case
simplified
Meyer-Brandis (2004)
BN-S model. The utility-indifference price in case of continuous asset process with
independent stochastic volatility has also been studied. However, the most advanced
result, achieved by Becherer (2001), allowed the volatility process only to switch
discussed the MEMM in the
between
In
case
a
of the
finite number of states.
of the true BN-S
model,
neither the MEMM
nor
utility-indifference pricing
has been solved.
We
can
summarize the status of
existing results
as
follows:
Indifference
MEMM
Pricing
Measure
Additive Processes
Continuous
Asset, Independent Volatility
BN-S Models
Contribution and
1.3
(Yes)
No
Yes
(Yes)
No
No
Organization
of this Thesis
This thesis widens the range of results in two dimensions:
•
On
one
hand,
we
identify
contingent claims
the
written
optimal martingale
on
measure
which may be described
as
indifference
utility
pricing
additive processes.
of
no
measure
of
European
analysis of the
contingent claims
in
case
This extends the
contingent claim
functions of the value of
to the set of
some
risky
asset at time T. We
generalize this result by introducing a second risky asset, which is assumed to be
the only tradable asset, i.e. we consider the case where some contingent claim is
defined on some asset which is not traded, however, another asset may be used to
hedge the contingent claim. Such a situation can arise in electricity or insurance
markets. The result of this problem is closely related to the first one.
•
On the other
including
hand,
the MEMM for
the
the
The
class of jump-driven stochastic
the BN-S model is identified.
model that the asset process itself
literature
a
on
stochastic
utility indifference
volatility process.
can
Important
to note is that
jump. Hence, this is
a side product,
models. As
volatility
pricing measure
for
a
volatility
we
models
allow in
our
true extension in the
we
may also determine
European contingent claims
written
on
corresponding techniques are very similar. We determine a defining equation for the
optimal martingale measure, which we develop from a representation result of Grandits
and Rheinländer (2002). This equation, analyzed for the first time in a Levy process settting, provides an inspired guess for the shape of the optimal martingale measure. With
this approach, the optimal martingale measure may be determined by the solution of a
boundary problem which includes a nonlinear partial integro-differential equation. One
additional important result of this thesis comes from the analysis of nonlinear partial
CHAPTER 1.
6
INTRODUCTION
differential equations. We show existence and uniqueness of solution in
second order
Let
us
In the
case
of first and
problems.
present the organization of the thesis.
general semimartingale framework, Chapter 2 defines the concepts
of utility indifference pricing and hedging and applies them to the exponential utility
function. For this purpose, the important duality result by Delbaen et al. (2002) is sum¬
marized. The main focus is then put on the dual problem, i.e. the identification of the
optimal martingale measure. By a modification of a very important characterization re¬
sult by Grandits and Rheinländer (2002), we identify necessary and sufficient conditions
for the optimal martingale measure. Based on these conditions, which ensure that some
potential martingale measure truly is the optimal martingale measure, we determine a
so-called verification procedure, which will be widely used in the following chapters.
In
setting of
Chapter 3,
a
we
by introducing
start
the
filtration,
which is
generated by some Levy
predictable representation
integral with respect to the
Due to this technical construction and the 'weak
process Y.
every local
property',
continuous
martingale
part Yc and
a
is the
stochastic
sum
integral
of
a
stochastic
with respect to ßy
—
vy, whereas ßy is the
of the discontinuous part of Y and vy is its compensator.
jump
of adapted, locally bounded assets, the class of all equivalent martingale
measure
For
a
measures
model
is dis¬
by Grandits and Rheinländer (2002), we then develop
optimal martingale measure. This equation has a rather com¬
defining
plicated structure, such that no explicit solution can directly be drawn from the equation.
However, applied to the case of additive processes with no contingent claim, one easily can
conjecture an inspired guess of the form of the MEMM by this equation. This inspired
guess needs only to pass the verification procedure. Due to its importance, we call it the
'Optimal Martingale Measure Equation'.
cussed. Due to the characterization
equation for the
a
Chapter
4 has the form of
ferential
equations",
we
a
an
special
intermezzo.
The topic is so-called "interactive
class of nonlinear
partial
and 3, the importance of
reason
chapter
to dedicate for it
Becherer and Schweizer
(2004),
dif¬
differential equations, for which
discuss existence and uniqueness of classical solutions.
the main
partial
Not motivated
by Chapter
2
4 will be unfolded in
a
our
Chapter 5 and 6, which is also
separate chapter. Primarily inspired from ideas from
proofs are based on the Picard-iteration technique.
Having summarized the main ingredients, we are able to attack the two problems. In
Chapter 5, we discuss the problem of utility indifference pricing in case of additive pro¬
We work directly in the set-up with two assets, where the European contingent
cesses.
claim is defined
auxiliary function u,
dependent
log-return
By the 'Optimal
Martingale Measure Equation', we conjecture the form of the auxiliary function u, which
results in an interactive partial differential equation. By results of Chapter 4, we can show
that the auxiliary function exists and is uniquely defined. By the verification procedure
as presented in Section 2, we can then prove that the measure, defined by the auxiliary
function u, truly is the optimal martingale measure. We will also show that the one asset
which is
the non-tradable asset.
on
on
time and the
We introduce
some
of the nontradable asset.
CONTRIBUTION AND ORGANIZATION OF THIS THESIS
1.3.
closely
case
is
By
similar
a
related to the two asset
argumentation
as
for the
7
case.
problem
of
utility
also determine the MEMM in
indifference
pricing
in the additive
of stochastic
volatility processes,
Differently to the case before, the auxiliary function u de¬
pends on the level of stochastic volatility Vt. A general result can be provided for the case
of bounded stochastic volatility. As this condition is not fulfilled in case of BN-S type
models, we extend the results for BN-S type models to unbounded volatility processes.
As a by-product, the MEMM in case of a 'continuous asset process with independent
stochastic volatility' is also treated.
process case,
we can
which is done in
The results
2005b).
Chapter
presented
case
6.
in this thesis
are
summarized in Rheinländer and
Steiger (2005a,
8
CHAPTER 1.
INTRODUCTION
Chapter
2
General Results for
Exponential
Utility
Merton's classical
By
trade in
a
a
stock is to allocate money
terminal wealth is maximized.
the
function,
optimal strategy
gale
measure.
The
same
by
his
after
trader may
own
so
It is well known that in
goal
of
that his
case
an
investor who
can
from
expected utility
exponential utility
of the
linked to the so-called minimal entropy martin¬
closely
is
expected utility by entering into the market
derivative and investing his initial wealth
addition, issuing
premium. Following Hodges and Neuberger (1989), the utility indif¬
try
now
to maximize his
account or, in
collecting
the
portfolio optimization problem,
risk-free bond and
the
a
ference price of the claim is then defined
as
the premium for which the investor becomes
indifferent between the two investment alternatives.
ference price, the
utility
indifference
Consistently with the utility indif¬
hedging strategy is the optimal adjustment of the
strategy for the investor who has issued the claim.
portfolio
In this chapter,
nential
utility
we
introduce first
maximization
(2002).
Rheinländer
by
some
notation and summarize
Delbaen et al.
We then discuss in
(2002),
some more
hedging and an approximation result is presented.
to identify the optimal martingale measure.
2.1
Frittelli
detail
duality
(2000)
utility
results
expo¬
and Grandits and
indifference
We close with
on
some
pricing and
auxiliary
results
General Semi martingale Framework
probability space (Q, T,P), a finite time hori¬
zon
a
(J-"t)o<t<T satisfying the usual conditions of right-continuity
and completeness. For simplicity, we assume that Tq is trivial and Tt
T. All semimartingales are taken to have right continuous paths with left limits. Expectations are
taken with respect to P unless specified otherwise.
The mathematical framework is
T and
filtration F
given by
a
=
=
Let S be
a
risky
an
E-valued
asset in
constant at
a
(P, F)-semimartingale.
financial market which contains
1, and
always
assume
are
now
a
as
the discounted price of
risk-less asset with discounted price
that
S is
We
We consider S
F-locally
bounded.
confronted with the question about the fair price of
9
a
contingent claim
CHAPTER 2.
10
B
well
as
GENERAL RESULTS FOR EXPONENTIAL UTILITY
how the appropriate
as
systematically,
let
us assume
that
hedging strategy looks like. To approach the problem
our overall objective is to maximize the expected utility
of wealth, i.e.
f
supE U(x+
u(x+
where
x
+
J0
etdSt-B)j,
(2.1.1)
at time T under
trading strategy 9 and initial
exponential type:
-*))]
9tdSt is the portfolio value
endowment x, and U is
our
which is assumed to be of
utility function,
An investor
Definition 2.1.1
OtdSt-
is
assumed to have
U(x)
=
exponential utility function, if
an
--e~ax
a
with
In
some a >
addition,
the
to
0, called the risk
existence of
ensure
parameter.
aversion
a
solution,
we
consider
contingent claims which
are
of
form:
following
Definition 2.1.2
We
with B
define
a
of Tt-measurable contingent
subset
claims
B,
which
fulfill
In
a
first step,
E
e(a+e)B
we
want to
optimal strategy.
price
Let
as
us
well
In
as an
a
a
martingale
the linear
is
=
E
martingale
measure:
subspace of L°°(Q,Jr,P), spanned by
integrals of the form f
h(ST2
such that the stopped process ST2 is
f
0.
investigate the maximal expected utility and the corresponding
we will show how these two results provide a fair option
recall first the notion of
random variable. A
all
some e >
later step,
stochastic
for
for
< oo
appropriate hedging strategy.
Definition 2.1.3 V
times
e~eB
E
and
< oo
measure is
—
Stx),
where 0 < T\ < T2 < T
bounded and h
a
the
probability
is
bounded
a
measure
Q
elementary
are
stopping
T^-measurable
<ti P with
E[jpf]
=
0
V.
We denote
of all
of
by M. the set
probability measures
which
martingale measures and by M.e the subset of M. consisting
Here and in the sequel, we identify
are equivalent to P.
measures with their densities. Note that, as S is locally bounded, a probability measure
Q absolutely continuous to P is in M. if and only if S is a local Q-martingale.
A very important concept in case of exponential utility functions is relative entropy.
Definition 2.1.4
to the
probability
The relative
measure
R
is
entropy I(Q,R) of the probability
defined
I(Q,R)
'
is
well known that
Csiszar
(1975)
I(Q, R)
> 0
with respect to
Q
with respect
as
Er
dQ lncr dQ
if
dRiu& dR
and that
Q
<
R,
otherwise
-foo,
It
measure
I(Q, R)
further properties of
=
0
if
and
only if Q
the relative
entropy.
=
R.
We
refer
to
GENERAL SEMIMARTIN GALE FRAMEWORK
2.1.
Let
of martingale
also introduce the space
us
with respect to
some
measure
case
To be
of
R=
P,
{Q
:=
usually only
we
measures, which have
finite
relative
entropy
R:
Mf(R)
In
11
write
g
I I(Q,R)
M
<
oo}.
M.f.
ought to specify the class of all 'permitted' trading strategies 6.
However,
doing this because Delbaen et al. (2002) showed that there are
several possible choices which lead to the same maximal expected utility. Technically, 6
will be a subset in the space L(S) of predictable and »S-integrable processes. Let us now
summarize the most important results in case of exponential utility optimization:
precise,
we
also
we
refrain from
Theorem 2.1.5
Given B
B.
e
We
assume
that S
locally
is
bounded and 6
is
a
set
of
'suitable' strategies. Hence:
1.
Solution
of
Dual Problem
a
It holds
sup£"
exp <
—
alx
—
+
see
(
inf
QeMf V
=
2.
Existence
If
QB
G
exp <
—
ax
+
Eq [aB] -i(Q,P)
(2.1.2)
Solution
of
there exists
measure
-
[ 6tdSt-B)}
a probability measure Q
G M.?
M.? C\ M.e, which maximizes
M.e,
C\
then there exists
3.
all
Q
G
Mf.
Characterization
We consider
and
only if,
unique
(2.1.3)
EQ[aB]-I(Q,P)
over
a
a measure
the
Solution
of
Q
following
G
M.?
dP
a
constant
4>B
G
QM
=
cB
\0
E
as
M.e.
Then, Q
the
is
maximizer
QB of (2.1.3), if
holds:
dQ
for
C\
well
L(S)
exp<[>+ /
-
as
an
(f>fdSt
+
aB\
(2.1.4)
jo
S-mtegrable predictable
f 9dS
is
a
(Q, F)
-
process
martingale for
(pB,
all
Q
G
Mf.}.
equality (2.1.2) expresses that instead of solving the primal problem
of maximizing expected utility over all strategies, one can consider a dual problem, which
For the dual
is minimizing a functional over a set of appropriate martingale measures.
have
well
known
and
characterization
be seen by
existence
we
as
can
problem,
results,
2. and 3. Hence, we will concentrate m the following on the dual problem.
Occasionally, we will require that 6 contains ®m It ensures that the optimal investment
Remark 2.1.6
strategy
is
The
attained
m
6.
CHAPTER 2.
12
Proof: The
duality (2.1.2)
introduce the
us
GENERAL RESULTS FOR EXPONENTIAL UTILITY
has been shown
by
1
'
easily
shown that for all
HQ.P)
However, by assumption,
G
M.f,
—
EQ[t\B\]
a
suitable
>
e
have that
+
I(Q,P) corresponds
know that there exists
we
<
I(Q, P)
+
-E[e^}
I(Q,Pb),
if,
and
•
which
3. is
(2002), taking
a
M.f
C\
I(Q,Pb).
M.e. Hence,
we
(2.1.5)
< oo
Ll(Q)
well
as
as
Q
G
=
since
—
direct consequence of
into account that
Q
G
M.f
3.2 of Grandits and
Proposition
C\
M.e is the minimizer of
I(Q, Pb)
only if,
there is
and
c
Jo (f)BdSt
•
QB,
provided.
is
Rheinländer
a measure
Q
G
Mf(Ps)
Mf(P). By
and
existence
Q G Mf(Pß) n Me,
(2002),
minimizes I(Q, Pb) respectively maximizes Eq[aB]
0 and therefore B G
Theorem 2.1 and Remark 2.1 of Frittelli
uniqueness of
to the minimizer of
a measure
(2002),
have due to Lemma 3.5 of Delbaen et al.
for
we
+
Eg[aB]
maximizer of
Q
^[logil+log^j+aB
HQ,PB)
log^s- EQ[aB].
=
=
Hence, the
let
E[eaB]
dP
be
showing 2.,
aB
.=
can
For
Pb via
measure
dPß
It
(2002).
Delbaen et al.
(pB
g
>,
as
an
.S-integrable predictable
well
process
<f>B,
such that
^ß-
=
exp
< c
+
as
e^-
Hence,
dQ
dQ dPB
dP
dPB dP
=
=
^aBi i
exp{c-log£[0+
exp{cs+ / 4>fdSt
Jo
with cB
:
=
c
-
/
/
+
iB,
4>BdSt
+
aB}
aB}
log E [eaB].
Notation 2.1.7
minimal entropy
The
measure
martingale
Q°
minimizes
measure.
I(Q, P) overQ
G
M.f
and
is
therefore
called
UTILITY INDIFFERENCE PRICING AND HEDGING
2.2.
Let
Indifference
Pricing
B;a)
exp <
Utility
2.2
us
—
sup£"
:=
eee
Definition 2.2.1
//
there
is
=
u(x
+
a(x
—
tts(B)
/
J0
+
9tdSt
tts(B; a)
=
—
B)
>
.
J J
to the
equation
ns(B)-B;a),
utility indifference (selling)
call this solution the
price
(2.2.1)
for
B.
adjustment to the initial capital that compensates an
investor for the additional terminal liability B in terms of maximal expected (exponential)
utility. In this sense, tts(B) is a subjective fair valuation of the liability B from the
perspective of a risk averse investor with exponential utility function.
can
be
—
LI
unique solution
a
u(x;a)
7TS(B)
Hedging
introduce
u(x
we
and
13
interpreted
Remark 2.2.2
the
as
Analogously,
one
can
u(x; a)
also
=
u(x
define
—
utility indifference (buying)
the
nb(B)
B; a).
+
by 7rb(B)
the
Furthermore,
utility indifference (selling) price of
who already has a liability C is given by
It
is
easy to
that the solution
see
is
given
—tts(—B) if
=
ns(B\C)=ns(B
+
an
Let
us now
Corollary
is
explicitely
2.2.3
well-defined
C)
-
u(x
=
u(x-ns(C)+ns(B
+
C)-(B
=
u(x
+
C)).
describe the
+
7rs(B;a)
for
defined.
an issuer
the
exponential utility
7TS(C))
ns(B\C)-(B
utility
Under the assumptions
and given
is
C)-ns(C),
=
-
the latter
additional claim B
if the terms of the right-hand side are defined. To see this, note that
implies that tts is not dependent on the initial capital x. Hence,
u(x
price:
+
C))
indifference price:
of
Theorem
2.1.5, the utility indifference
price
as
=
=
=
!Eq[B]--(i(Q,P)-I(Q°,P))}
sup
l
QMf
a
V
J}
(2.2.2)
EQB[B]-^I(QB,P)-I(Q°,P))
(2.2.3)
-(c°-cB).
(2.2.4)
a
Proof: Existence of
rectly
from the
Q°
and
QB
follows from Theorem 2.1.5.2. We get
defining equation (2.2.1), taking
equality (2.2.2)
into account the definition of
u(x—B, a)
di¬
as
CHAPTER 2.
14
well
as
GENERAL RESULTS FOR EXPONENTIAL UTILITY
equality (2.1.2). Representation (2.2.3)
We know from Theorem 2.1.5.3 that
Let
EQB[aB]
=
gies and B
cB,
G
Assume S
2.2.4
L°°(P).
(2000):
Then
locally bounded, M.f C\J\4e ^ 0,
is
a i—
tts(B; a),
a
liimrs(B;a)
(0, oo);
G
of
strate¬
and
EQo[B].
=
a{0
Aie, Proposition
suitable set
QeMe
limirs(B;a)
is in
a
non-decreasing
is
EQ[B],
sup
=
«î°°
price of
+
c°.
=
1.3.4 in Becherer
Proposition
cite
Proposition
Q°
from Theorem 2.1.5.2.
immediately get representation (2.2.4).
we
us now
Since
directly
may write
we
I(QB,P)
I(Q°,P)
Hence,
also follows
particular that the utility indifference
interval of possible arbitrage free valuations,
implies
2.2.4
any bounded claim lies within the
in
that is
inf
Utility
indifference
justment of
strategy that
Let
is
us assume
<
hedging
ns(B;a)
can
be
<
EQ[B]
sup
for B
deduced from
easily
utility
L°°(P).
G
indifference
We
that the representation of the
dQ
\
=
dP
exp
„B
/
,
•; c
optimal martingale
iBjo
/
-\-
cpt aot
measure
,„ol
>
-\- ar>
is known.
Corollary
2.2.6
The
strategy 9B
sup£"
—
—^
:=
exp <
is
(x
a
—
an
+
optimizer for
/
OtdSt
B) >
—
see
We
therefore get
i){B)
Proof:
pricing:
define the utility indifference hedging strategy ip(B,a)
optimal martingale strategy without liability that is necessary
optimal under the terminal liability B.
Definition 2.2.5
the
EQ[B]
Due to
equality (2.1.2),
supE
we
=
i){B-a)
exp{—a(x
-e~ax
=
—e
exp{
-ax—c
9B -9°.
may write
see
=
=
+
/
Jo
9tdSt
sup
Q£Mff\Me
—
B)}
1
(EQ[aB]-I(Q,PJ)\
as
the ad¬
to obtain
a
ZERO MARGINAL UTILITY PRICE
2.3.
Let
us now
consider 9B.
Hence,
we
get
/
exp{-o;(x+
E
15
efdSt-B)}
therefore, 9B truly
Zero
2.3
Note that the
is
an
tïs needs not be
concept of
Price
a
linear price function.
since due to diversification
subjective pricing,
two risky claims is in general not
approximated price function may
Let
twice the value of
ue(x,B;a)
u(x
:=
+ tts (e)
supE{U(x
—
claim.
subjective value of
However, a linearly
a)
eB ;
irs(e)
+
+
eee
ns(e) corresponds
to the
We know that for every
Hence,
we
du(x,B;a)
us now
Then,
e
fedS-eB)],
J
utility indifference price for the contingent claim of size
fixed, there exists the corresponding optimal strategy 9e.
may write
=
de
Let
the
be introduced.
=
eB.
effects,
risky
one
This is consistent with the
consider the function
us
where
dP
optimizer.
Marginal Utility
price
B
—e
=
and
dQ
-E
=
consider the
e\u'(x + 7Ts(e) + [6dS eß) ((vr^e))'
-
'principle
of
equi-marginal utility',
the price function is defined in such
impact the investor's utility function. Hence,
(O'OO
E[U'(x
E[U'(x
we
+
B
ß~eue(x,B; a)
changes
equals
zero.
in volume do not
get
/ 9dS eB)B]
7Ts(e) + j9"dS-eB)}
ns(e)
+
=
i.e.
way that small
a
-
+
-
such that
(ns)'(e)de
ns(B)
(O'OO
We
e=0
get
(O'OO
E
e=0
where 9° is the
f9°dS) B
E[U'(x + j9°dS)]
U'(x
optimal investment strategy
utility function, the pricing measure
_
case
of
no
claims. In
j9°dS)
E[U'(x + j9°dS)]
U'(x
dQ
dP
in
+
+
(2.3.1)
case
of the
exponential
CHAPTER 2.
16
to the
corresponds
MEMM,
U'(x
GENERAL RESULTS FOR EXPONENTIAL UTILITY
since
/ 9°dS)
+
=
{
exp
a(x
-
/ 9°dS)
+
with
E
a(x
exp
/ 9°dS)
+
>
exp{—ax
=
—
c0},
such that
/ 9°dS)
E[U'(x + j9°dS)]
U'(x
+
exp
< c
o
(f)0dS
Remark 2.3.1
1.
2.
In economics,
this valuation
(1871).
In
Kallsen
(2001) identified
who all
system
nously
financial mathematics,
Let
measure.
us
a
it
is
long history dating back to devons
analyzed by Davis (1997) and Foldes (2000).
a
where the MEMM
approach,
consisting of
market
expected utility of
where, however, the price processes of
maximize
,
another
consider
has
principle
their
a
number
terminal wealth.
given. Since any contingent claim that
derivative traders behave
identically,
we
is
the natural pricing
of
identical investors
We consider
a
closed
the
underlying securities are exogebought has to be sold and since the
with the following market clearing
is
end up
condition:
Derivatives prices should be such that the
sentative investor contains
He then states that the
no
E
price
fulfills
/ 9°dS) B
E[U'(x j9°dS)]
U'(x
+
+
Verification Procedure
2.4
(2002)
As stated in Theorem 2.1.5, Grandits and Rheinländer
describe
measure
us
measure
first find
some
candidate
us now
1:
J0 (f)f dSt
measure
Q,
which
can
rT
'
exp|ci?+
dP
Step
a
criterion for
to coincide with the
dQ
Let
identified
a
optimal martingale measure. Let us therefore
a procedure,
consisting of three steps, for verifying that a given probability
the
equals
optimal martingale measure:
martingale
Let
the repre¬
contingent claim.
corresponding
V
optimal portfolio of
-
/
(pfdSt
+
be
represented
as
aß\
jo
describe the Verification Procedure:
Q
is
+ oiB
an
>
equivalent probability
is
integrable
E
measure, i.e.
we
have to show that exp
with
B
exp
<cr
+
4>f dSt
+ oiB
=
1.
\
cB
VERIFICATION PROCEDURE
2.4.
Step
2:
Step
3:
Q
is
a
The
measure, i.e. the traded asset is
martingale
probability
measure
I(Q, P)
Step
f (pB dS
4:
is
If all these conditions
For
carrying
out
are
Step 1,
=
Eq[cb + / <j)f dSt + aB]
local
Q-martingale.
P,
i.e.
< oo.
Jo
for all
is the
fulfilled, Q
will need the
we
a
has finite relative entropy with respect to
Q
Q-martingale
true
a
17
Q
G
M.e with finite relative entropy.
optimal martingale
following
measure
result which is
QB.
generalization
a
of the
Novikov condition for discontinuous processes:
Lemma 2.4.1
Let N be
a
locally
bounded local
P-martingale.
Let
Q
be
a measure
defined
by
§
dP
where AN
>
—
//
1.
Ut
has
a
=Zt
=
\{Nc)t
+
predictable compensator Bt
Q
is
J]{(1 + ANS)log(l
This result is
a
ANS)
+
as
well
J' r(p dS
if, for
be
a
some
>
an
(2.4.2)
measure.
equivalent martingale
Q-martingale.
local
ß
be
0,
This is due to the
exp <
(2.4.1)
<oo,
direct consequence of Theorem III. 1 of
LetQ
ANS}
as
Lepingle
Finally, to cope with Step 4 of the verification procedure,
presented in Rheinländer (2005), Proposition 3.2:
Lemma 2.4.2
-
s<t
equivalent probability
an
S(N)t,
the process
E[<sx$BT]
then
=
Tt
Then
J' rtp dS
ß fQ ip^d[S]t f
is
is
measure
a
we
with
and Mémin
mention
finite
a
(1978).
result,
relative
which is
entropy. Let
(square-mtegrable) Q-martmgale
true
P-mtegrable.
inequality
r-T
ßEQ
/
$d[S]t <I(Q,P)
'0
which follows from
f îfttdSt
is
a
true
(2.1.5),
and
Q-martingale.
+ -E
exp
/3 /
e
Q-integrability
$d[S]t
+1
< oo,
10
of the
quadratic
variation
ensures
that
18
CHAPTER 2.
GENERAL RESULTS FOR EXPONENTIAL UTILITY
Chapter
3
Martingale Properties of
Jump-Diffusion Models
2 shows the
duality between optimal strategies and martingale measures. Let
For this purpose,
us now start to discuss and identify the optimal martingale measure.
we consider a specific class of asset models, which we call jump-diffusion models.
They
are characterized as being stochastic processes adapted to a filtration which is generated
by a Levy process. This condition is rather technical, however, several well known and
frequently used models fall under this assumption. We will further see that due to the
weak predictable representation property, this assumption is also very useful from a math¬
Chapter
ematical point of view.
Under
of
additional structural condition
an
equivalent martingale
measures
on
well
as
as
the asset process,
the so-called
we
will describe the set
Optimal Martingale
Measure
equation.
We will conclude with the
Martingale
Measure
presentation of
equation
can
be
a
prominent example showing how the Optimal
applied.
Filtration and Asset Model
3.1
that the filtration F is the
complete
natural filtration of
Let
us
Y.
We refer to
Appendix
and
uY(dx,dt)
martingale
v{dx)dt
corresponding compensator, respectively.
assume
and the
measure
(Yc)t
=
=
t. Let
us now
Let
predictable a-field
To any measurable
,
s
on
introduce
short introduction to
us
Q
define Ù := Q
x R+ and B(R)
function
W
on
Ù
Levy
processes.
part of the process
simplicity,
For
Let
Y,
process
Yc,
its
ßY
jump
we assume
that
we
x
x
is
R+
x
R and V
the Borel
:=
V ®
B(R),
where V
is
the
field ofR.
associate the process
dx)
if
J \W(oj,t,x)\uY({t}
x
dx)
< oo,
otherwise.
+00
by Qioc(ßY)
Levy
notation:
some
J f W(oj,t,x)uY({t}
[
We denote
a
be the continuous
Notation 3.1.1
-
A for
a
the set
of
V-measurable real-valued
19
functions
W
on
Ù such that
CHAPTER 3.
20
the process
MARTINGALE PROPERTIES OF JUMP-DIFFUSION MODELS
Wt, defined
as
Wt(uj)
\/^2s< (Ws)2
satisfies
By
G
Afoc,
it has
i.e.
locally mtegrable
variation.
Shiryaev (1987) (abbreviated JS in the following),
following weak predictable representation property:
Jacod and
have the
Theorem 3.1.2
Every (P,¥)-local martingale
M
for
/ W(cü,t,x)ßy({t},dx)-Wt(uj),
:=
some
H G
We therefore
respect
L2loc{Yc),
[ HdYc
Mo+
+
can
W(x)
*
be written
(ßy
an
we
as
Giocißv).
G
write any local
a
III.4.34,
Vy)
-
martingale M as the sum of a
vY.
integral with respect to ßy
F-adapted, locally bounded semimartingale, it has
can
to Yc and
Since S is
W
=
M
Theorem
stochastic
stochastic
integral
with
—
the canonical
decompo¬
sition
S
where M is
with
By
locally
a
locally
the weak
bounded local
finite variation.
Since
=
So
martingale with M0
S is locally bounded,
=
predictable representation property,
M
=
=
where Mc and Md
are
Mc +
(1992),
we
0 and A
it is
write M
a
predictable process
special semimartingale.
a
as
Md
f aMdYc + WM{x)
*
{ßy
-
Vy)
the continuous and the discontinuous part of the local
M, respectively, aM is predictable and WM
and Strieker
A,
+ M +
G
Gioc(ßY)-
As
already
martingale
by Ansel
shown
that the finite variation part A of the
absence of
arbitrage implies
absolutely continuous with respect to the predictable quadratic
variation process (M) of the martingale part M. We therefore assume that the asset price
process S satisfies the following
asset process S must be
Assumption
3.1.3
(Structure Condition)
There exists
a
predictable
fying
At=
[
Jo
Xsd{M)s,
with
(T
KT:= I
Jo
\2sd{M)s
< oo
P-a.s.
process X satis¬
CLASS OF
3.2.
EQUIVALENT MARTINGALE MEASURES
Class of
3.2
Equivalent Martingale
21
Measures
probability measure Q on the filtered probability space (Q, J7, F, P) is called
an equivalent martingale measure, if it is equivalent to the physical probability P and
if, under Q, the discounted price process S is a local martingale. M.e is the set of all
equivalent martingale measures.
Recall that
a
Lemma 3.2.1
Doléans-Dade
Q
Let
M.e.
G
exponential
as
Proof:
well
Let
as
us
[M, L] being
first note that
=
local
we
S(
-
/
S +
we
-^
is
given
by
the
may write
=
S(N)t
martingale. Due to the Structure Condition, we know by JS,
that J XdM is a locally square-integrable local martingale. Hence, let us
local martingale
We have to show that
P,
:=
XdM + L
L:=N+
unter
Zt
local
being a
Theorem 1.4.40,
introduce the
process
P-martingales.
Zt
with N
density
process
Z
with L
the
Then,
get due
f -^-d[S, Z]
is
a
[M, L]
is
a
local
[ XdM.
martingale.
to Girsanov's Theorem that S is
local
P-martingale (see
He et al.
Since S is
a
special semimartingale
Q-martingale if, and only if,
local
(1992),
a
Theorem
XII.12.18).
Let
us
write
S+
By
f --)-d[S,Z]
=
M+
=
M +
=
M+
J Xd(M)+ J --)-d[S,l+
Xd(M)
+
[M
+
/
f Z_dN]
Xd{M},- /
XdM +
I'Xd{M)- f Xd[M] + [M,L] + [f Xd{M),- f XdM + L\.
inequality, we know that f |A|<i(M) and
Aioc- Hence, by JS, Theorem 1.3.18, we conclude that
the Kunita-Watanabe
belongs
to
L]
f Xd[M]
-
f X(M)
=
J Xd([M]
-
therefore also
f |A|<i[M]
(M))
martingale. On the other hand, by JS, Proposition 1.4.49, [f Xd(M), f XdM +
L] is a local martingale. Therefore, [M, L] must be a local martingale to ensure that
S + f -g-d[S, Z] is a local martingale.
is
a
Let
local
us
note that
measure:
—
we can use
the above representation to
ensure
that
Q
is
a
true
martingale
CHAPTER 3.
22
Corollary
by
MARTINGALE PROPERTIES OF JUMP-DIFFUSION MODELS
Let
3.2.2
the Doléans-Dade
Q
probability
a
exponential
Z
with L
Proof:
proven
well
as
Q
measure
is
as
a
that SZ is
f Xd{M)
[
The first three terms
local
are
-
a
local
a
local
martingale
martingale
sufficient to
have
as we
due to
L is
ensure
Due to Theorem
P-martingale. However,
this
can
be
f Xd{M), f Z_d(- f XdM + L)].
[
+
since the
P-martingales
integrands
are
locally
bounded.
we
of Lemma 3.2.1.
and WL
us
=
the fifth term
of the
is
a
density process is
martingale measure.
G
(3.2.2)
Qloc{ßy).
introduce
write the
Z
Finally,
1.4.49.
I<7LdYc + WL(x)*(ßY-isY),
Wz(x)
can
proof
may write
a
we
in the
(M)
-
locally bounded, the representation (3.2.1)
corresponding probability measure Q
3.1.2,
Llc(Yc)
seen
XdUM]
that the
Remark 3.2.3 Let
such that
probability
(Z_dM
JS, Proposition
L=
G
the
holds for the fourth term since
Hence, whenever
with aL
(3.2.1)
P-martingales. Then,
local
a
+
(M))
/
is
given
t Z_d(- t XdM + L)\
M,
+
t Z_Xd([M]
-
is
is
^p
IS_dZ+ ! Z_dM+ f Z_d[M,L]
=
same
:=
[S, Z]
I' S-dZ + (Z_Xd{M)
+
Zt
XdM + L)
bounded local
l S-dZ + f Z-dS +
=
process
measure.
only have to show
by product integration:
=
The
S(- /
=
We
SZ
density
process
[M, L] being locally
martingale
whose
measure
density
Aa
:=
a
:=
WL(x)
—
process Z
S(J (TZdY°
+
=
,
-
XWM(x)
(Zt)
(WZ(x)
-
+
1,
as
1)
*
(ßy
-
Vy)\
.
OPTIMAL MARTINGALE MEASURE
3.3.
EQUATION
23
The process
Ytc- / vfds
Jo
is
a
Q-Brownian
motion and
Vy(co, dt, dx)
is
of
the compensator
represents the risk
of
Wz(x)
—
1
be
can
interpreted
the process
martingale part
az
Yc and the
the risk premium associated with the
as
ofY.
the discontinuous part
Optimal Martingale
3.3
III.3.24). Hence,
Theorem
premium associated with the continuous
predictable function
jumps
Q (see JS,
under
ßy
(u), t, x)uy(dt, dx)
W
=
Measure
Equation
equivalent martingale measures and the criterion (2.1.4)
our starting point for identifying the optimal martingale
The characterization of the set of
equation, which
provides
an
measure
QB:
Theorem 3.3.1
strategy (p and the
The
r-T
aB +
c
b
\{oB
I
,
is
Jo
\aff
-
constant
+
c
(2.1.4) satisfy
m
<PBAt(af )2
+
the equat ion
f(WtM(x))Mdx)
tf\t
dt
r-T
AtKM)^c
((wL(x) (<PB X)WM(x))
^L-(0f
+
10
+
+
where the
+
-
(Ylog(l
predictable
-
XWM(x)
process
aL
G
*
WL(x))
+
Lfoc (Yc)
(ßy
+
as
uy))T
-
XWM(x)
well
as
WL(x)) ßY)
*
-
WL
G
(3.3.1)
Gioc(ßy)
have to be chosen
[0, T].
(3.3.2)
such that
°¥°t
Proof:
Let
be written
us
as a
+
consider the
stochastic
/ WtM(x)WtL(x)u(dx)
density
as
well
as
process
exponential
Z
with L
=
Vt
0
G
Jr
[M, L] being
local
=
^-
,
which due to Lemma 3.2.1
can
of the form
S(-
=
Zt
f XdM + L)
P-martingales. By
the representation property,
we
may write
[M, L]=
f afaLds + WM(x)WL(x)
Jo
We further know from Dellacherie and
Meyer (1980), VII.39,
*
ßy.
that the
process
(M, L)=
f afaLds + WM(x)WL(x)
Jo
*
vy
predictable
bracket
CHAPTER 3.
24
MARTINGALE PROPERTIES OF JUMP-DIFFUSION MODELS
locally bounded. However, (M, L)
martingale. Therefore, we get that
exists,
since M is
d(M,L)t
erfoLt
=
dt
consider
us now
log Zt
equal
to
+
-
log Zs_
-
s<t
XsdMs
+
is
a
local
Jr
^s
pt
[M, L]
the Itô Lemma:
log Z by applying
10
since
\ WtM(x)WtL(x)u(dx)
+
f -^dZs -l-f -^d(Zc)s Y,(log Z.
=
zero
0.
=
Let
is
Ï
Lt-\
~^^ZS)
[ Xsd(Mc,Lc)s-\(Lc)t
^""c
Xid(Mc)s+
'o
5>s II
zs_
/
A
XdM
ALS)
-
s<t
t
(^
On the other
-
1
Xsaf)dYsc
+
((WL(X) XWM(X))
+
(Ylog(l
-
due to equation
hand,
logZT
XWM(x)
-
ca +
=
+
(ßy
*
-
WL(x))
(2.1.4),
B,
<f>fdSt +
\
(Xso-f
--Jq
we
-
o-^Yds
Uy))t
+
XWM(x)
-
WL(x)) ßy)
*
.
may write
aB
T
cB
=
<pfa^dYtc+(<pBWM(x)*(ßy-uY))T
+ aB+
Jo
+
We
/T(VfAt(af)2
get equation (3.3.1) by combining the
Corollary
tions
are
3.3.2
Equation (3.3.1)
m
+
0fAt
j(WtM(x)fu(dx))dt.
equations introduced above.
two
Theorem 3.3.1
is
fulfilled
once
the
following
condi¬
satisfied:
i)
\WL(x)
n)
It holds:
(<PB
-
X)WM(x)\
+
*
ßY G
Aioc,
T
aB +
c
M\2
B
K-wr
WtL(x)
aL
+
-
{$
+
-
(<Pf
+
,
+
iB\
/
M\2
#%(o
Xt)WtM(x)
+
dt
<pfXt(WtM(x))2ju(dx)dt
Xt)^)dYtc
U\og(l-XWM(x) + WL(x))-(t>BWM(x)\
*hy)
•
(3-3.3)
EXAMPLE: THE MEMM IN THE ADDITIVE PROCESS CASE
3.4.
i) implies according
Proof: Condition
(WL(X)
-
((J)B
X)WM(X))
+
*
(ßy
-
to
JS, Proposition
Vy)
II.
(3.3.1)
this into account, equation
we can
write
(wL(x)-((J)B + X)WM(x))*ßY
=
~(WL(X)
Taking
1.28, that
25
reduces to the
-
((J)B
+
X)WM(X))
*
Vy.
simpler equation (3.3.3).
(2005)
(2004), V)ho perform an analysis m a pure Brownian motion set-up. In this
equality (3.3.1) is comparable to equation (3.4) m Rheinländer (2005). However,
Remark 3.3.3
The ideas
presented
m
this section
are
taken
from
Rheinländer
and Hobson
sense,
due to the jumps considered
m
our
case, the
corresponding problem
becomes much
more
demanding.
The MEMM in the Additive Process
Example:
3.4
Case
The aim of this section is to illustrate the usefulness of the
equation.
return is
We consider the
an
no
contingent
additive process.
claim case, i.e. B
Optimal Martingale
=
0, and
We therefore
a
(MEMM)
Chan
(see e.g.
(1999), Kallsen (2000), Miyahara (2001), Fujiwara
Esche and Schweizer (2005)). Let us specify our asset process by
-^
We will work with the
Assumption
•
ft(x),
at
Measure
that the asset
investigate the minimal entropy martingale
slight generalization of the well-studied case of Levy processes
in
measure
assume
=
ritdt
following
+
crtdYtc
\^f(x)
+ d
restrictions
on
*
(ßY
-
isY)J
the parameters
(tj,
and
Miyahara (2003),
.
a,
/):
3.4.1
and rjt
being deterministic,
bounded functions with
/ ff(x)u(dx)
(3.4.1)
< oo.
Jr
•
<Jt > cr* > 0
for all t
In the notation of Section
G
[0,T]
3.1,
we
At
and
some
write
=
af1
this into account, equation
r \(aL
c°
-
(3.3.3)
St-Xtot?
+
[ (wtL(x)
Jr
v
-
WtM(x)
JvJ?(x)v(dx)
can
+
lo
o
St-at,
=
V-
7
St-{o-2t
Taking
constant a*.
St-(4
be written
=
St-ft(x)
as
well
as
(3.4.2)
^.
as
Sl4Xta2)dt
+
K)ft(x)
+
jQ {oL-St-(4 ^dYtc
+
((log(l
+
Sl^Xtft2(x))u(dx)dt
7
+
-
S_Xf(x)
+
WL(x))
-
S_ctPf(x)) ßy)T.
*
(3.4.3)
CHAPTER 3.
26
One
directly
constant.
if
we
MARTINGALE PROPERTIES OF JUMP-DIFFUSION MODELS
that
sees
By (3.4.2),
that
assume
(p
must choose
we
A
S_X is
:=
S-(p°, aL
'=
terms with random elements
WL(x)
Replacing aL
for
WL(x)
and
such that the RHS of
WL(x)
Hence, the RHS
deterministic functions
are
as
(3.4.3)
is
is constant
well
as
both
i.e.
=
($+X)a,
=
Xf(x)-l
(3.4.4)
we
+
exp($f(x)}.
(3.4.5)
immediately get
the
following
condition
<p>:
0
a2($+X)
=
r\ +
=
Hence,
a
strategy
(p°
candidate. We will
o2(p
+ X
+
verify
[0,T]
4>(t),
R with
(pt
:=
Vt +
the
addition,
f(x)(exp{(f)f(x)}
suppose
which
°% +
martingale
corresponding cp
Proof: Let
(3.4.6).
the
us
ensure
For this purpose,
measure
measure
a
potential
is in fact the MEMM:
-
v
there exists
l)is(dx)
function (p
a
(3.4.6)
0.
=
'
y
Q, defined by
f
exp
\c
o
T i
,
+
<-
normalizing constant,
first
fulfills the above equation is
/ ft(x)(exp{$tft(x)}
Jr
dP
being
l)v(dx).
Assumption 3.4-1- Then,
solves for any t G [0,T]
dQ
with c°
—
'
^
Jr
that the candidate
us
—
/
[ f(x)v(dx)+ [ f(x)(exp{$f(x)}-l)is(dx)
Jr
for which the
Theorem 3.4.2 Let
In
and
equation (3.3.2),
in
<p>°
and
deterministic function.
a
vanish,
aL
WL, aL
is
I
Jo
,c
1
-^—dbt >,
j
St-
the MEMM.
that there exists
a
bounded function
cp
which solves equation
introduce two functions gi,g2 with
we
9i
92
(p
!->
<Tt<f>
rit + tri
,
~
-//^(expW.M)-!)^).
:
Obviously, g\ is continuous, strictly increasing with gi(—co)
other hand, #2 is continuous, #2(0)
0, and decreasing, since
=
—oo,
#1(00)
=
00.
On the
=
g'2(4>)
Hence,
we
can
#2 is
=
f2(x)e^p{(f)f1(x)}iy(dx)
-
<
0.
positive for negative values of (p and negative for positive values of (p. Hence,
conclude that there exists
a
value
(ptl
such that
g\((pt)
=
9i(4>t)-
In
addition, cpt
EXAMPLE: THE MEMM IN THE ADDITIVE PROCESS CASE
3.4.
\(pt\
is bounded since
g\. We conclude that
is well-defined. We
Q
The local
respectively.
(i) of Corollary
,
must be smaller than
martingale
the absolute value of the null of function
now
define aL and WL
given by (3.2.2).
L is then
27
Let
by (3.4.4)
us
and
(3.4.5),
check that condition
3.3.2 is satisfied. We have
WL(x)
(f
-
+
X)WM(x)\
*
ßY
exp{0/(x)}
=
1
-
(Pf(x)
-
ßY
We know that
exp{#(x)}
1
-
-
(pf(x)
*
Vy <
OO
since
exp{0/(x)}-l-0/(x) <(<Pf(x))
in
a
neighborhood
of
/ (x)
=
0, together with (3.4.1) and the boundedness of (p.
We
therefore conclude that
exp{0/(x)}
the solution
Therefore,
also
a
(pf(x)
-
*
Ajoc.
ßy G
(c°, 0°, aL, WL) given by (3.4.4), (3.4.5), (3.4.6)
(3.3.1).
solution to
1
-
In the
following,
we
perform
(3.4.3)
is
the Verification Procedure
as
and
outlined in Section 2.4:
1.
Q
is
an
equivalent probability
measure:
We will check the conditions of Lemma 2.4.1. Let
N defined
XdM + L
:--
4>adYc
exp{0/(x)}
+
-
1
equality follows from (3.2.2) together
bounded, N is locally bounded and AN >
where the last
/
are
martingale
as
N
and
consider the local
us
)
*
(ßy
with
—
-
vY).
(3.4.7)
(3.4.4), (3.4.5).
Since
(p
1. We have further to show
that
U
=
-
/
(p2a2 ds
+
\(pf(x) ey+){(pf(x)}
-
exp{(pf(x)}
+ 1
)
*
ßy
locally integrable variation as well as its compensator fulfills (2.4.2). Since
locally bounded, locally integrable variation follows directly by proving that
has
if(x) exp{4>f(x)}
-
has finite variation. For this purpose, let
g(z)
It
can
be
easily
shown that
=
g(z)
z2
—
> 0
exp{$f(x)}
for all
I)
analyze
us
zexp
+
z
z
+ expz
< 1:
—
1.
*
ßy
U is
CHAPTER 3.
28
MARTINGALE PROPERTIES OF JUMP-DIFFUSION MODELS
g reaches
0 and Z2
=
=
Further, g'(z)
1-
have therefore two local extrema z\
we
convexity
g(z)
In
for z\
zero
for all
> 0
addition,
at the
zexpz
extrema,
get g"(z\)
we
>
0, and therefore,
neighborhood of f(x)
the
of
hand,
U, defined as
compensator
a
=
B
is
Q
\ S^V
is
Therefore, by
martingale
a
Since WL is
3.
In 2.
l|
<
($f(x))2
U has finite variation.
i^f{x) exp{^/(;r)}
ds +
+
"
exPW/(x)}
+
l)
On the other
*
iv
Hence, the condition (2.4.2)
2.4.1, Q is an equivalent measure.
well defined deterministic function.
a
fulfilled.
2.
:=
=
Hence,
expz + 1 > 0.
J/(x)exp{J/(x)}-exp{J/(x)}
in
zexpz and
Analysing the
g"(z2) < 0. Therefore,
0 and Z2
0 and
—
< 1.
z
—
=
2z
=
to
I(Q, P)
< oo:
The
Corollary
density
Z
I(Q,P)
=
=
naturally
measure:
bounded,
get due
Lemma
is
[M, L]
the processes L and
3.2.2 that
Q
is
be written
jp may
EAé
St-
o
martingale
a
as
<
exp
are
bounded.
locally
Hence,
we
measure.
c° +
f0 0° dSt
Hence,
>.
dSt
T
EQ
Let
us
denote Vy
J (psa2ds
are
=
cu+
/
<j>t [ritdt
ex_p{(pf(x)}
Q-martingales:
we
mdt
:=
mt
:=
*
+
atdYtc )
f(x)
Then
vY.
*
Uf(x)
+
(ßy
—
*
Vy)
(ßY
as
-
well
vY)
as
f0 asdY^
have that
(jUy-iv)j
/(z)*
o„dY?
o
are
local
P-martingales (even
,t
d
m,
1
'o Zs
-d{Z,mu'}s
=
true
P-martingales). Hence, by
(jUy-Iv)j
/s(x)exp{0s/s(x)}
-
/(x)
/O
^s
-d(Z,mc),
(
+
fs(x))v(dx)ds
Jr
*
fs(x) exp{4>sfs(x)}v(dx)ds,
ßy
t
=
theorem,
fix)*
[f(x)*(ßY-vY)i
Jo
m,
Girsanov's
JO
t
asdYsc
-
(psasds
EXAMPLE: THE MEMM IN THE ADDITIVE PROCESS CASE
both local
Q-martingales. In fact, they
quadratic variations are Q-integrable (due the
are
as
condition
(3.4.1)).
(pf(x)
We conclude that
29
Q-martingales since their
boundedness of /, a and (p as well
true
are
(ßy
*
Vy)
—
is
Q-martingale.
true
a
Further, since
4>f(x)*(v^ -Vy))
we
< oo,
T
may write
(pf(x) *(ßY-Vy))
EQ
(pf(x)
=
T.
*
(l#
,Q
Vy)
-
r-T
/
4>t
ft(x)(exp{(ptft(x)}
l)v(dx)
-
dt
T
<M
~'f]t-
o](pt\dt.
On the other hand,
rT
rT
i2o2tdt.
(pt(JtdYtc
EQ
Therefore,
T
E:
Q
St-
Hence,
.
we
f 4j-dS
is
(pt(rnt dt
E,
dSt
Q
I(Q, P)
true
Q-martingale
(pf(x)
+ I
v
o
have proven that
a
dYtc)
+ at
for all
Let
us
bounded and
locally
(ßy
-
vY)
0.
/T
is finite.
Q
G
M.e with finite relative entropy:
We will check the condition of Lemma 2.4.2. Since S_ is
zero, -g- is
*
therefore,
f -§-dS
is
locally
local
a
bounded away from
Q-martingale.
discuss
rT
E
On
one
hand, exp{ J0 a2dt}
from He et al.
E
exp
(1992),
is deterministic and
(3.4.1)
*
ßy
we
/ s~dS
hand,
we
know
exp
-
Jr
'0
<^ (f(x))
*
/,
we
l)v(dx)dt.
(3.4.8)
conclude that
ßy
get
E
and
e{ft{x))2
T
and the boundedness of
E
is finite. Hence
On the other
finite,
Lemma 14.39.1. that
<^ (f(x))
Due to condition
a2dt+((f(x))2*ßy
exp
exp
is therefore
finite relative entropy.
by
a2dt+[(f(x))2*ßy
Lemma 2.4.2
a
true
< OO
T
Q-martingale
for all
Q
G
M.e with
CHAPTER 3.
30
MARTINGALE PROPERTIES OF JUMP-DIFFUSION MODELS
We conclude that
^
dQ
fulfills all sufficient conditions for
f
being
o
,
fT
$t ,c\
the MEMM
Q°.
Remark 3.4.3
1.
(3.4.6) corresponds to a well known condition for the MEMM m case of
Levy processes (e.g. condition (C) m Fujiwara and Miyahara (2003), or condition
(4-4) m Theorem B m Esche and Schweizer (2005)). Differently to these papers, we
Condition
assumed local boundedness
to other papers,
generally for
2.
we
of
the asset process.
determined the MEMM not
On the other hand and
only for Levy
differently
processes but
more
additive processes.
of a nonvanishmg continuous martingale part
uniformly bounded. However, this condition can
is sufficient to assume that
The restriction
has been used to
(p
difficulties.
be weakened without
sure
that
is
It
o\
is
bounded away
from
zero.
+
/ ft(xHdx)
Jr
en¬
Chapter
4
Interactive Partial Differential
Equations
In the
previous chapter,
since this
equation
tion of the
6,
we
will
we
identified the
optimal martingale
d
—u(t,z)
:=
u(t,»), A
nonlinear in ut,
g is
a
is in
measure
play
a
is
a
+
(cB, <pB, aL, WL),
general
an
than first-order PDE's and
an
the identifica¬
easy task. In Sections 5 and
=
0,
we
gz(t,ut)
:=
Optimal Martingale
g(t,ut)(z),
Measure. Since
the expression interactive PDE's.
use
This
of existence and uniqueness of classical solutions of
analysis
not that well
In the first case,
Second-order PDE's
we
assume
In
a
analytically more amenable
are treated in many textbooks.
However, interactive PDE's
studied. Assuming Holder continuity of g, we will
inspired
-
(2004)
Ansatz of Becherer and Schweizer
problem.
that A is
often
are
-
show that there exists
a
unique classical
solution to such interactive PDE's. The Banach Fixed Point Theorem is the
solution of the
is
with PDE's of the above type.
uniformly elliptic operator.
general
the
identifying
We discuss two classes of interactive PDE's.
by
not
differential operator and g, with
function of the entire function ut,
chapter
boundary problems
in
equation. However,
parameters
(Atu)(z)+gz(t,ut)
crucial role for
is dedicated to the
are
Measure
that PDE's of the form
see
where ut
Optimal Martingale
consists of four unknown
addition,
will also
provide
key
for the
Feynman-Kac representation
of the solution. In the second case, A is assumed to be of first order. Applying again the
Picard-iteration technique, we will be able to prove existence of a weak solution. However,
differently
ensured.
we
a
to the second-order case, smoothness of the solution will not be
A classical solution
can
Section 4.1 treats the second-order
only
PDE,
be assured
Section 4.2
further
by
provides
restricting
the
analysis
automatically
the function g.
of the first-order
PDE.
4.1
Let
us
Solutions for Parabolic
start with spaces of Holder continuous functions.
Definition 4.1.1 For
which
Equations
Dktp
is
m
G
bounded and
N0;
let
C(R)
uniformly
consist
continuous
31
of
on
all those
R
for
functions
0 < k <
m.
tp G
Cm(R), for
CHAPTER 4.
32
Putting 0 < a < 1, r
for which Dm'p satisfies
a
+ a,
m
=
exists
INTERACTIVE PARTIAL DIFFERENTIAL
mR
define C£(R)
we
a
Banach space with
IIMIIr
||
Spaces
a
r
=
functions
those
m,
—
<p
that is, there
constant K such that
is
where
subspace of
of exponent
Holder condition
a
\Dm>p(x)
Cl(R)
to be the
EQUATIONS
||oo
•
is
-
Dm>p(y)\
-
norm
the supremum-norm
M
of Holder continuous functions
y\r~m.
-
\Dm>p(x)
SUP
+
(see
K\x
by
given
^Ilnfc
„
2^ WD ^ll~
<
|T
_
Dm>p(y)\
>
v\r-m
(1978), Example 1.27).
Adams
are a
-
natural
ential equations. This becomes obvious when
object in the theory of partial differ¬
discussing fundamental solutions of partial
differential equations of second order. Let
1
2#
8
CtU=2atd?U
be
an
elliptic operator
and at and bt
are
Definition 4.1.2 A
fundamental solution of
is a function T(s,Ç;t,z)
defined for all (t,z)
satisfies the following condition:
function f
btd~zU
[0,T]. Let
on [0,T].
R for each t G
on
continuous functions
For any continuous
+
the
us
assume
that
a
parabolic operator Ct
and
(s,£)
m
[0,T]
with compact support, the
is
strictly positive
+
J^
R with
x
m
s
[0,T]
>
x
R
t, which
function
Jr(s,Ç;t,z)f(OdÇ
u(t,z):=
satisfies
Ctu
+
—
u
u(t,z)
(1975),
We know from Friedman
0,
=
at
f(z) ift\
->
Theorem
s.
6.4.5, that there
T which is differentiable in time and twice
continuously
exists
a
fundamental solution
differentiable in space, with
D?r(8,&t,z)\<C(8-t)-^ew{-c^f}
for
m
=
0,1 and suitable positive
Lemma 4.1.3 For any
sup
I
r
'
constants
(1, 2),
G
dz
V
)
S»
)
we
)
/
C,
c.
But
we can
say
even more:
have
Qz
^s,iU,,)-iT(s,iU,y)\diiC{s_t)_i
y
\z
y&+y+z Jr
—
for
some
suitable constant C
(4.1.1)
SOLUTIONS FOR PARABOLIC
4.1.
The constant C is
depending
EQUATIONS
33
of the lower bound of
Taylor (1996), Chapter 15). The proofs
sophisticated instruments. However, within our
usually
setting, simpler techniques may be applied and for that reason, we present the proof of
are
(see
General results of this kind
at.
well known in the PDE literature
in the standard textbooks
e.g.
need
this theorem.
Proof:
direct
By
calculation,
"s
1
us
(£
f
z
-
exp
-
equals
fts budu)2
// o2udu
2
analyze
-3/2
JR
h(e)
symmetry
we
z
-
exp
2
show that for any k
we can
X"
xexp
(x
k
e)
+
budu)
fts buduf
-
// a2adu
0,
>
we
have
(x+t)2
exp
dx
3-r-
<
:
lr-1
we
reasons,
start
(ï-z
„du
(£
x
The claim is proven if
a
2v^
dz
Before
-1/2
\-V2
a2 du
d_ r(s,£;t,z)
For
^
/To,
2,R
Let
show that the fundamental solution
easily
one can
consider the
0
case e >
proving inequality (4.1.2), let
only.
analyze
us
(4.1.2)
Ck^.
some
integrals.
For
a,b ER,
we
have
x
x
x2exp
As
a
exp
=
xexp
dx
(x2
=
h(e),
first approximation to
h(t)
<
2
x
hand,
exp
<
>
—
(x
X
<
L
<
exp
r
a
4
/
~k
^
x2
r
x
2kel~r
exp <
l
—
k
dxt
dxt
\-r
\-r
--•9i(e)-
may write
we
——
dx
x
exp
x
/ \x\
Jr
Jo
=
~k
k)
+
x
exp
get
we
=
On the other
~k
x
dx
X
x3exp
x
dx
xexp
x[ exp
+
e)
exp
2
~~k
exp
(x
+
e)2
k
(x
+
k
e)2
(4.1.3)
dx
>)
dx +
t
I
exp <
(x
+
tf
>dx.
CHAPTER 4.
34
Let
us
discuss the first term in detail. Let
(x
exp
+
e)2
>
exp
>
exp
>
exp
k
(x
exp
we
INTERACTIVE PARTIAL DIFFERENTIAL
+
e)2
k
us assume
x
2ex
~k
~k~
x2
2ex
T
~k~~~k
x2
2ex
~k
~k~~~k
(x
x
exp
~~k
x/
exP
i
<
x >
since
(for
e
>
(4.1.4)
2'
for 0 <
(4.1.5)
<
x
exp
exp
consider the
(x
+
e)2^
case x <
+
e)2
[
for
k
2exA
x >
exp
<
exp
k
for 0 <
k
(again
0. Since
for
>
e
x
<
0)
-2
-
<
k
(x
ef
"
2
us now
+
x
utX
t
Zj,Xj
,
,
for
~~k~~~k^ ~~k~\
T
x2
2ex
~k
~k~
for
k
x <
<x<0,
get
(—x) (
(x
exp
+
tf
X'
exp
2ex
exp
+
k
<
exp
On the other
hand,
(—x) (
due to the
(x
exp
~k
=2
x2
\
2ax2
k
\
k
inequalities (4.1.4), (4.1.5),
+
e)2^
(~x)
exp
x[
x <
,
exp
we
may write
x
~k
x2
1
2 ex
k
\
k
for
x <
for
x <
k
<
exp
We conclude that for
< x< 0.
-2-
-
exp
k
x <
for
0
>
for
f(^-¥)
k
0,
x2
we
ê_
2ex
_
k
k
k
for
|
for
k
< x <
0,e
<
-2x,
< x <
0,e
>
-2x.
have
(x
+
tf
i
k
+
~k(X
2ex
2
k
for
<?x
k
for
0)
2x2
(1
e2
I 2nx/
k
exp
we
for
e2
Hence,
0.
x >
get
0 < x\ exp
Let
that
EQUATIONS
x <
< x< 0.
2'
SOLUTIONS FOR PARABOLIC
4.1.
We therefore
EQUATIONS
35
get
(x
x
xi
exp
exp
~~k
2e
x
x2exp
<
k
tf
+
dx
dx
k
r2e
2
k
X
e
-x
k
2e
exp
X
exp
e2
0
.2
k
X
x
k
X
l_ _2e2_
2e
-x
kx
2e
x
<
x
x
x
by
the
x
x3exp
/
Jr
inequality (4.1.3),
to
exp
<
——
dx
dx
we can
\dx
Since g\ and g2
\fïtkl'2.
=
k J
\-
approximate
h(t) <2vW/2e2"r
<7i(e) Ag2(e),
x
xexp
results from above and
integral
Hence, due
dx
dx
xexp
k2
x
exp
dx
xexp
2e2
3
k*x
dx
dx
exp
'
dx
+
e3"r
=:
g2(e).
strictly decreasing respectively increasing on e
bounded by ^i(eo), whereas e0 is given by #i(e0)
are
h is
>
=
0,
as
well
#2^0),
as
i-e.
h(t)
<
eo must
fulfill
2V5Ffc1/2eg-r
+
e33-r
2/ce0
,
respectively
7r)k 1/2
2(V>7T
eo
We conclude that
fr(e)
and
therefore,
For any
e
Cb'r([0,T']
>
x
22-r(V^T2-V^)1~rk^L
<
the claim is proven.
0,
we
E),
fix T'
:=
T
—
e.
For
r
G
which consists of functions
(1, 2], let us introduce the set of functions
:
[0,T;] xR^I, which are continuous
u
CHAPTER 4.
36
INTERACTIVE PARTIAL DIFFERENTIAL
[0/T'] x E and for
Cb'r([0,T'] x E), equipped with
in
(t,z)
\\\u\\\r
[0,T'] fixed,
any t G
G
the
a
Let
u(t, )
G
Crb(R).
The space
d
\u(t,z)\+
\—u(t,z)\
sup
(t,z)£[0,T']xR
VZ
l-§-zu(t,x) --§-zu(t,y)\
sup
sup
t[0,T] x,y&R,x^y
is
:=
norm
sup
(t,z)£[0,T']xR
:=
ut
EQUATIONS
\X
y\
—
Banach space.
us
[0,T]
now
x
introduce
Cl(R)
->
he
some
C6(E),
C£(E).
G
for which
We further consider
continuous function g
a
:
the notation
we use
gy(t,ut) :=g(t,ut)(y).
For any function
?0,r
G
u
x
C6'r([0,T/]
(Feu)(t,z)
E),
=
let
introduce
us
integral equation
an
of the form
[r(T't;t,z)he(Od£
Jr
rT'
T(s,Ç;t,z)gi(s,us)dÇds.
We will show that under suitable conditions
E)
for
some
appropriate
Lemma 4.1.4 Let
functions
Cb'r(R)
with
there exists
a
is
C61,r(E)
such that
constant L <
oo
any
r
\\\u\\\ß
(1, 2), Fe
G
+
for ß large enough.
Remark 4.1.5
sup
t£[0,T]
In
This
point
m
however,
a
we
still suitable
Cr(R)
D
a
e
-ß(T'-t)
Pl
;
Cb'r([0, T'] x
<
+
on
G
to
our
purposes.
The
[0, T']
E)
x
i.e.
(4.1.6)
with respect to the
norm
e"^71'"*^— u(t, z)\
VZ
rjr-j
y\
—
unique
fixed point
Proposition
2.1
ue G
as
well,
idea
of
Cb'r([0,T']
x
E).
Becherer and Schweizer
m
of
then showed that
of u
conceptual
need to control the derivative
:
Q^u(t,y)\
—
:
Cb(R), they
that g
-v'2\\oc)-
stochastic representation
a
continuous
d
\X
a
x
||^i
sup
(t,z)£[0,T']xR
\d^U\t)X)
has then
as
m
Cb'r([0,T']
—
of
Cb'r(R), uniformly m t,
[0,T"]; v\,V2 G Cb'r(R), we have
L[\\vi -W2II00
for
sup
x,y£R,x^y
similar result
have chosen
for
in
the set
is
us now assume
G
all t
e~ß{T'~t)\u(t,z)\+
boundary he belongs
Cb- Since
C£{R)
v
contraction
particular, Fe
is
(2004). However, they
suming that the
is
sup
(t,z)£{0,T']xR
:=
:=
function
continuous
such that
\\g(t,vi) -5f(t,^2)IU
Then, for
unique fixed point
a
(1,2).
Lipschitz
a
Fe has
g,
Holder continuous derivatives. Let
bounded,
Cb(R)
—
G
(1, 2]
G
r
r
on
our
the
As¬
the operator
Fe.
Fe has
fixed
general,
result
proof
a
is
is
unique
less
the
same.
4.1.
SOLUTIONS FOR PARABOLIC
EQUATIONS
37
equivalent to the norm 11| || |r, we know that (Cb'r([0, T'\ x
E), HI \\\ß) is a Banach space. In addition, due to the estimates of (4.1.1) as well as
Lemma 4.1.3, we know that Pe is an operator from Cb'r([0,T'] x E) into itself. Hence,
Proof: Since the
11| || \ß
norm
is
•
•
•
existence and
uniqueness of
contraction
Cb'r([0,T']
Due to
on
\\g(t,uht)
we
have for U\,u2
g(t,U2j)Woo
-
<
Cb'r([0,T']
G
L(\\ultt-U2,t\\0o
x
+
sup
x;,y£R,x;^y
Taking
inequality (4.1.1)
into account the
e-Kr-V\(FeUl)(t,z)
r(s,£; t, z)
eß(T'-t)
for
eß(T'-t)
CL
/TT
tt2
-
I
Let
us now
and
discuss
=
0,
we
C(s
|| |/3
m
ni
z
G
\u'i,t{x)
-
u'2tt{y)\\
—
\X
y
may write
/
—
t)
R(T'-s)
e«
2
\x-C\2-
exp
s-t
d^'^ds
i
>ds
E.
T^(Feu)(t,z).
Since
rT'
C(s
—
1
t)
\z-i\-
exp
/(s,mm)
s-t
rT'
<
\\u'l,t-U2,t\\oo
HF
,CL
[0,T']
m
E)
-
<-^T7^\J-\\\ui-u2\\\ßJt
for all te
show that Pe is
(gt(s,uit8) g*(s,u2,s))dÇds
L
\U\
can
(Feu2)(t,z)\
-
1
<
we
Ë).
x
inequality (4.1.6),
fixed point will be proven if
a
CL\\\u\
u21| \ß /
—
(s
—
t)
1
-
gt(s,u2,s)\
\z-t?-
exp
s-t
dÇds
e^'^d^ds
r-T'
CLj-\\\Ul-u2\\^ /
and
by
Holder's
inequality, taking
p <
2,
(s-t)-l/2e^T'-^ds,
-
+
-
=
1,
T'
(s
-
t)-l'2e^T'-s^ds
rT'
.
1/p
.
t-T'
(s-t)~p/2ds) (/ e^T's)ds^
<
1/9
1/9
qi/ißi
r-T'
<
ß1/«
1/p
1
(s-t)-^2ds)l/P'e^\
V1
a
CHAPTER 4.
38
we
directly
see
INTERACTIVE PARTIAL DIFFERENTIAL
that
P~ß(T'-t)
—
(FeUl)(t,z)-—(Feu2)(t,z)
1
d_
eß(T'--t)
dz
the
d_ r(s,£;t,z)
eß(T' -t)
CL
<
T(s,Ç;t,z) gt(s,ui>s)
r-T'
1
<
By exactly
ß1/qV
same
r-T'
h
c,Jt
reasoning,
\
(s
-
e"^'-*)
-
1
-^y^|ll^i
£FeUl(t,y)
ÈFMfx)
~
+
£Feu2(f
\x-y\
Jt
I
I dz
—
T(s,C;t,x)-i-T(s,!=;t,
I
1^—1
\x
x>yeR>x^y>
r-T'
RlT,_fy\\\ui
eP{±
d£ds
r/2|| 1/3-
—
—
y\
x|/(s,mm)
<
gt(s,u2>s)\
lr-1
rT'
eß(T'-t)
-
finally get
we
^ FMfx)
sup
x,y£R,xj^y
i/p
t)~p/2dsj
gt(s,u2>s) dÇds
-
\gt(s,ui>s)
dz
9
<
EQUATIONS
''
~u2\\\ß
I
sup
x,y£R,x^y,
Jt
-
gî(s,u2tS)\d^>ds
f\fr(s,C;t,z)-§-zr(s,C;t,y
\z-y
r-\
-di
xe^T-^ds
<
^S^\\\Ul
eß(T'-t)
T {s-tyh^-'^ds
U2\\\ß
-
I
rT
<
for
some
CiLf
ß1^ (/
vjt
(s-t) ?2 ds)
suitable constants
C\,
We conclude that there exists
p <
a
ß
IH-FVui
and Fe is therefore
Let
us now
a
us
>
z
Q
=
u2\\\ß
1-
0 such that
FeU2\\\ß
-
<
respect
consider the linear
coefficients, defined
boundary condition
d
dt
for which,
P
contraction with
(Af)(z)
tion with
and è +
-
-
\\\ui
to the
-
we
u(t, z)
shall
+
on
|| |/3
|||
•
H^.
case:
partial differential operator
bt^J(fz) °l^f(fz)
[0,T].
+
Consider the semi-linear
partial differential
equa¬
at time T:
(Atu) (z)
assume:
=
tt2
norm
state the main result for the second-order
Theorem 4.1.6 Let
with real
-[jj\\\ui
+
gz(t, ut)
=
0
u(T, z)
=
h(z)
for
all
for
(t, z)
all
z
G
G
(0, T)
E,
x
(4.1.7)
(4.1.8)
SOLUTIONS FOR PARABOLIC
4.1.
coefficients b and a2
bounded away from zero.
The
a-1
EQUATIONS
G
r
uniformly
is
(t,z)
bounded.
gz(t,v)
i—
is
For
(1,2],
G
r
Holder continuous
m z
G
v
[0,T],
m
function
m t.
uniformly
Cb'r(R),
the continuous
i.e.
—
a-3 g
functions
bounded continuous
are
(1,2], condition (4.1.6) is fulfilled,
Cb(R) is Lipschitz continuous m
Cb'r(R)
a-2 For
39
fix a function v
uniformly with respect
g
(t,z)
to
is
[0,T]
:
Cb'r(R).
G
we
a2
and
x
Then,
compact
m
subsets.
a~4 h
Then,
:
E
there
(4.1.8),
E
—
Lipschitz
is
continuous and bounded.
unique classical solution
is a
Cb([0,T) xE)
G
u
C;([0,T) xR)
andCb ([0,T)
E)
x
=
Cb'\[0,T) xE)nC1>2((0,T)
of bounded,
the set
is
continuous space derivatives
[0,T)
on
continuous
E.
x
/
u(t,z)=
T(T,!=;t,z)h(OdC+
Jr
where V
is
fundamental
the
We fix
C1,r([0,T]
x
E)
some
d
boundary
Let
Existence and
+
/
Jt
Jr,
At
a
=
solution
:=
as
§^
m
[0,T]
function
u
(4.1.9)
E.
x
Cb ([0,T)
G
x
E)
n
=
0
for all
(t,z)
G
(0,T)
E
x
h(z)
w
G
VzgE.
C1,2((0,T)
are
x
E)
follows from Friedman
the solution is
given by w(t, z)
=
[r(T,!=;t,z)h(OdC
/
Jt
by
bounded,
Jr
+
—
xE with
be written
condition
(Fu)(t,z)
w(t, z)
+
some
(1975), Theorem 6.4.6 and Corollary 6.4.2. In addition,
(Fu)(t, z) with operator F being defined as
well as,
[0,T]
on
can
E)
it
T(s,!=;t, z)g^(s,us)d!=ds,
start with
(Atw)(z)+gz(t,ut)
uniqueness of
Since h and g
u
x
r
/
the operator
us
w(T,z)
as
Cauchy problem (4.1.7)-
and consider the PDE
7rw(t,z)
with
(1,2).
G
r
of
solution
functions
The solution
t
Proof:
to the
where
/ r(s,f;t,z)0*(s,us)d£ds.
Jr,
bounded, for any t G [0,T) fixed, w(t,-) and j^w(t,-) are bounded
4.1.3, j^w(t,-) is uniformly Holder continuous. In addition, since
t | T and h is bounded and Lipschitz continuous, we conclude that
Lemma
h(z)
for
wgC6o,1([0,T) xE).
Let
e
<
us
fix
some
T and T'
:=
uo
T
G
—
Cb([0,T)
e.
By
the
x
E),
such that
we
define wo
:=
Fuo-
preceding discussion, wo(T', ) belongs
Further,
to
C£(E).
we
fix
Let
us
CHAPTER 4.
40
here introduce the
INTERACTIVE PARTIAL DIFFERENTIAL
auxiliary operator Fe,
defined
Cb'r([0,T']
the Banach space
on
EQUATIONS
x
E),
as
follows:
T'
/ T(T'&t,z)wo(T'Od£+ /
(Feu)(t,z)=
By
For
Lemma
eo
[0, T']
4.1.4, Fe has
fixed,
x
let
—
—
Let
e0-
E)
x
Let
us
fix
Cb'r([0,T']
ue G
(Mi)„>no
with no such that
a
—
Cauchy
^
<
the set
on
sequence in the
norm
e~ß<yT
~^\u(t,z)\
e"
sup
(t,z)£[0,T']xR
n0, such that
n > m >
we
-ß{T'-t]\lz^z)\
(t, z)
may write for
G
[0, T']
x
E:
u±(t,z)
n
rn
=
[ (V(T
-
-,C;t,z)w0(T
-,0 -T(T
-
-
-,C;t,z)w0(T
F(s,Ç;t,z)gt(s,ÛL<s)dÇds- /
and
E).
x
show that this sequence is
us
with
sup
(t,z)£[0,T']xR
:=
ß big enough.
ui(t, z)
T
:=
Cb ([0,T;]
I INI 1/3
and
consider the sequence
us
E with T'
Banach space
unique fixed point
a
/ T(s,&t,z)gt(s,u8)d£ds.
-,0)dC
-
r(s,Ç;t,z)gt(s,û±JdÇds,
therefore,
n
m
/r(T--,e;t,z)K(T--,e)-^o(T--,e)Me
<e-^'-*)
n
+e
f
-ß(T'-t)
n
m
]T{T_}_^.t^z)_T{T_L^.tjZ)llwo{T_L^m
n
m
m
rT-±
+e~ß(T'-t)
+e-/3(T'-t)
r(s, £; t, z)\g*(s, «i.)
/
conclude that there exists
a
similar
reasoning,
e-/3(T'-t)|
az
constant
a
e-ß(T'~t) 1^ ^ zj
By
"|^(s,«i>a)|r(a,e;*^Kds-
T(-, -,t,z),
Due to the boundedness of
/(s, ux „)|d£ds
-
_
^
we see
ui(t,Z)
n
1
1
m
n
<K,
JfT(-, -,t,z),
Xi,
^ Z)\<Kl
we
t|u>o
and g
1
CL
m
n
~fTy
a
on
[T"
+
y,T]
x
may write
1
that there exists
-
vüq,
such that
constant
c
\Ui
Ux.
—
n
m
\\\ß.
K2 such that
T-Ux(t,z)
az
m
CL
/^V
c
(s-t)^2ds) V-^\
\Ui
n
—
U±
rn
\\\ß.
E,
we
SOLUTIONS FOR PARABOLIC
4.1.
EQUATIONS
41
We conclude
d
^
7T«iI
oz
for
-
az
n
...
^
7T«i
<
/3
m
Wi
1-CL
/3 big enough. Hence,
we
,
-V/3
'
Kl
+
K2
w
rT
ßi"t
(ui)n>n0
is
a
1
m
n
In
+^(/;(^-i)-p/2^)i/p^
have shown that
1
Cauchy
70,1,
sequence in Cb
([0,
T'\ x
n
E) converging
to
a
Uo(t,z)
function «0
G
Cb ([0,T;]
x
E),
which fulfills
llmui(t,z)
=
n—>oo
«
r(T--,e;t^K(T--,eK
lim
n
ri—>oo
n
T-
F(s,Ç;t,z)gi(s,ûi JdÇds
rT
T(s, £; t, z)gi(s) û0,s)dÇds.
T(T,Ç;t,z)h(OdÇ
The order of limes and
integrals may be changed by the
uniformly bounded.
Since eo may be arbitrarily chosen, we conclude that u
unique fixed point for the original operator F.
since Wo and g
Let
as
us
well
For
show that
now
is twice
as u
showing
the first
u
G
Cb([0,T)
continuously
claim,
let
us
x
E),
G
Co,1([0,T)
have to show that
u
x
E)
is the
0,1/
G Cb
([0,T)
r(s,Ç;t,z)gi(s,ûs)dÇds
+
<
<
directly
d
dz
For the
conclude that
u(t, z)
analysis
||m||oo
dz
< oo.
(4.1.1).
Since h and g
/ C(s-t)-1/2exp
are
bounded
as
E)
well as, for
one
\z-C\
s-t
dÇ
C
On the other
T(T,Ç;t,z)h(OdÇ+
of the first term,
x
(4.1.10)
t,
\T(s,Ç;t,z)\dÇ
we
we
uo
consider
in connection with the estimations of
>
i.e.
=
differentiable in the space variable.
u(t,z)= / T(T,!=;t,z)h(0dC
s
dominated convergence theorem
are
easily
/
hand, by differentiation,
—
dz
get
/ T(s,bt,z)gi(s,û8)d£ds.
checks that
jyT(T^t,y) -^T(T^t,y).
=
we
CHAPTER 4.
42
since h is
Hence,
Lipschitz
INTERACTIVE PARTIAL DIFFERENTIAL
continuous with
Theorem that h is differentiable
Lipschitz
a
-irh
That is,
a.e.
constant
Lh,
know
we
is well-defined
EQUATIONS
by Rademacher's
and therefore, we
a.e.
may write
d_ T(T,C;t,z)h(0
d£
dz
d_ T(T,C;t,z) h(0
<
d£
[ \r(T,t;t,z)\\£-h(tM
<
Jr
°i
[\T(T,!=;t,z)\dC
Lh
<
=
LhC
Jr
For the second term, let
r-T
analyze
T
Q
r
us
dz
T(s,C;t,z) d£ds
<
C
<
C
(s
—
l
t)
d^ds
I exp
T
TV
C
Since g is
uniformly bounded,
we
7T
2C
<
(s
-
t)~l/2ds
Jt
(T-t)1'2.
conclude that also
'
dz
U\
< oo.
Hence, the first claim
is proven.
For
showing
the second
due to Lemma
auxiliary
d_
Again by
a
we
have
G
u
again
us
fix
C°bl([0}T'}
x
some
E)
n
T"
<
T.
C1,r([0,T']
By equation (4.1.10)
x
E).
Let
us
and
consider the
PDE
Jt
exists
4.1.3,
let
claim,
w(t,z)
Friedman
solution in
+
(Atw)(z)+gz(t,ut)
=
0
w(T', z)
=
u(T', z)
(1975),
Theorem 6.4.6 and
C1,2([0,T']
x
E),
arbitrarily chosen, we conclude that
Cauchy problem (4.1.7)-(4.1.8).
for all
(t,z)
for all
Corollary 6.4.2,
which
E
u
by uniqueness must
Cb([0,T) xR). Hence, m
G
(0,T')
G
z
is
E.
conclude that there
we
be
x
u.
a
Since T" may be
classical solution to
the
Differently to Becherer and Schweizer (2004), we assumed boundedness
of
coefficients m the differential operator. Therefore, classical results could be used for
showing existence and uniqueness of a classical solution to the Cauchy problem.
Remark 4.1.7
the
Let
us now
close
discuss
a
stochastic representation of the solution of the interactive PDE. The
theory of second-order differential equations and Markov
processes with continuous trajectories is well-known (see e.g. Freidlin (1985)). Sometimes,
probabilistic methods may play the role of a tool for deriving delicate analytical results,
as we
relationship
will
use
Corollary
in
between the
a
later step.
4.1.8 All
Assumptions a-l-a-4
are
fulfilled.
Let
us
consider the
diffusion
process
Zt
=z,
dZT
=
bsds
+
(jsdWs
(4.1.11)
EQUATIONS
SOLUTION FOR FIRST-ORDER
4.2.
with
a
standard Brownian motion W.
u(t,z)
This follows
by applying
Zls'z
(4.1.11).
We
get the well-known Feynman-Kac representation
h(Z^z)
E
=
43
/
+
gzts'z(s,us)ds
u(s,Zl'z)
Itô's formula to the function
is well defined and the
expectations
Zl'z
with
defined in
as
well defined since h and g
are
are
bounded.
Solution for First-Order
4.2
discussed the
Having
parabolic
case,
we
will
Equations
apply
will
similar
techniques
slightly change
the larger Banach
the set-up.
analyze
to
we
will
first-order
that g
Especially,
is Lipschitz continuous on
space of continuous functions, equipped with
the supremum-norm. This will reduce the complexity of the arguments. On the other
hand, differently to second-order equations, first-order equations do not have smoothing
effects generated by a diffusion term. Hence, there might not exist a classical solution.
To overcome this lack, we must introduce a broader notion of solution, a so-called 'weak
solution'. Roughly speaking, a weak solution may contain discontinuities, may not be
differentiable, and will require less smoothness to be considered a solution than a classical
equations. However,
we
assume
solution.
Let E be
an
intervall in E. Let
—u(t,z)
+
us
consider
b(t,z)—u(t,z)
boundary problem
a
of the type
c(t,z)
=
0
in(0,T)x£,
(4.2.1)
u(T,z)
=
h(z)
onE,
(4.2.2)
+
where
c, h
b,
are
continuous functions
b is
Since
b,
c,
h
Lipschitz
not assumed to be
are
on
[0,T]
x
E and
(4.2.3)
continuous.
differentiable,
classical solution is not ensured.
a
However, the problem (4.2.1) with boundary condition (4.2.2)
can
be understood in dis¬
tributions sense, that is
T
g
r
r
u(t,z)—cp(t,z)dzdt+
dt
JE
o
T
Q
r
/ h(z)cp(T,z)dz
,
.
rT
u(t,z)—(b(t,z)(p(t,z))dzdt+ /
Jo
Je
u^
v
y
(4.2.4)
Jo
/ c(t,z)<p(t,z)dzdt
=
0
Je
o
for test functions
<p>
G
Definition 4.2.1 A
C°°([0,T]
function
u
x
is
E)
with compact support in
said to be
a
weak solution to
(0,T]x
E-
(4.2.1)-(4.2.2) if
it solves
o
(4.2.4) for
Let
us
all test
functions <p>
G
C°°([0,T]
x
E)
further discuss the weak solution to the
with compact support
in
problem (4.2.1)-(4.2.2):
(0,T]x
E-
CHAPTER 4.
44
Lemma 4.2.2 Let
Zl'z,
which
which
we
there exists
Then,
m
E.
a
weak solution
of
the
that it
which may be written
Proof:
By
u, let
us
[0,T]
G
x
E,
let
us
introduce
(4.2.5)
z,
=
boundary problem (4.2.1)-(4.2.2)m C([0,T]
x
E),
as
u(t,z)
the Picard-Lindelöf
the initial value
Zi'z
b(s,Zl'z)ds,
=
stays
assume
(t,z)
holds. For
EQUATIONS
as
^-sZl'z
for
(4.2.3)
suppose that
us
defined
is
INTERACTIVE PARTIAL DIFFERENTIAL
problem (4.2.5)
motivate how
u
h(Zt/)+ f c(s}Zl'z)ds.
=
existence and
theorem,
is ensured. Before
uniqueness of
must look like. For this purpose, let
that all functions involved
are
U(s)
:=
sufficiently
u(t
+ s,
(t,z)
smooth. For
Zl'zs)
for
s
a
prove existence of
we
us assume
G
[0, T
G
-
[0,T]
x
solution
a
Zls'z
of
weak solution
that
E, let
u
exists and
us
define
t].
We may write
U(s)
d
=
=
such that
we
^^t
+
s,Zt'*8)
b(t
+
d
s,Zl£8)—u(t
+
+
s,z£*8)
-c(t-\-s,Z%8),
directly get
h(ZtTz)-u(t}z)
U(T-t)-U(0)=
=
[T-t
=
-
Jo
c(t
+
f
Jo
U(s)ds
s,z£*8)ds
r-T
c(s,Zl'z)ds,
't
and
so,
rT
u(t, z)
:=
7t,Z\
.
h(Z1/) +
/
/n
/
boundary problem (4.2.1)-(4.2.2).
setting of this lemma, let us show that
ryt,Z\
c(s, Zfs'z)ds
solves the
In the
Definition 4.2.1. For this purpose, let
//
:
E
—
E and its inverse
(//)_1
:
us
E
//(*)
:=
utr\v)
=
introduce
—
E
as
u
is the weak solution in the
some
follows:
J'b(u, f?(z))du,
ip-J'biutxr^du.
Z?
=
z
+
sense
of
notation. We define the functions
4.2.
SOLUTION FOR FIRST-ORDER
EQUATIONS
45
The inverse function is well-defined due to the existence and uniqueness of
the initial value
problem (4.2.5).
We will
the term
analyze
/
/ u(t,z) —(p(t,z)
(4.2.4). Analyzing
of expression
the part
+
—[b(t,z)(p(t,z)J
ft c(s, Zl'z)ds
=
a
solution of
dzdt
ft c(s, f^(z))ds
of
u(t, z),
get
T
T
p,
p,
c(s, fts(z)) \-J(t, z)
0
dt
JE Jt
+ —[
dz
b(t, z)4>(t, z)
dsdzdt
rT
c(s, fts(z)) I -4>(t, z)
+
Je Jo
o
t
c(s, <p)
g-zi b(f z)4>(f z)
dtdzds
[|^(t, (ftT1^)) l~z ( b(t, (ftvm^f (ftvm
+
d
x-^(ftT\<P)dtd<pd8.
For
d
g(t):=<f>(t,(f;)-\<p))^(ft>)-\<p),
let
us
consider
m
(^(t,(//)-l(^))^(//)-1^)
+(^,(/tr1(,)))|(/tr1(,)|(//)-1(,)
+Ht,(ftTl^))^(ftT1^)
=
Since
d
dt
we
(ftrL^)
=
b(t,(ftsr^))
have
d2
dtd'p
(ftT1^)
d
=
(^b{t,u;)-\*)))^u;r\<p)
and therefore,
git)
=
\jt<P(f (f?r\v)) + l~z (&(*, (ftrl(v))<p(f (Z/)-1^)))] ^(//)_1(v)-
We conclude
T
rT
'
rd
c(s, fts(z)) -6(t, z)
0
l0t
JEJt
/ c(s,'p) /
!
Je
o
T
'
o
Jo
g(t)dtdzds
f
/
o
d
+
Je
c(s,p:)g(s)dzds
/ c(s,'p)(p(s,'p)d'pds.
Je
,
—[
dz
b(t, z)4>(t, z)
dsdzdt
we
CHAPTER 4.
46
In
to the
analogy
"T
above,
INTERACTIVE PARTIAL DIFFERENTIAL
~d
r
.,
rT'd
=
/ h(<p) /
,..,_,,
u(t, z)
of
dzdt
d
„
[^(t,(//)-1(^)
JO
'E
h(ff(z))
-{b(t,z)(p(t,z))
+
,,
=
d
,
h(fir(z))[-(p(t,z)
JE
0
h(ZTz)
get for the part
we
EQUATIONS
+
^(6(t,(//)-1(^)0(t,(//)-1^)))
x-^(ft')-\V)dVdt
=
/
Je
Hence,
h((p)<f>(T,(p)d(p.
have shown that
we
(4.2.4) and, therefore,
Let
turn to
us now
is
our
We
which
we
use
again
b-2
Lipschitz
is
is
solution to equation
a
boundary problem (4.2.1)-(4.2.2).
class of interactive PDE's.
some
interval
m
+
R.
We consider the
partial differential
b(t,z)—u(t,z)+gz(t,ut)
=
0
m
u(T,z)
=
h(z)
onE,
the notations
gz(t, ut)
equa¬
g(t, ut)(z)
:=
well
as
(0,T)
as
x
(4.2.6)
E,
(4.2.7)
ut
:=
u(t, )
—
E.
continuous
m
:
E
continuous.
function g : [0,T]
Cb(E), uniformly m t, i.e. there
The continuous
G
v
\\g(t,vi)
further
alls
Then,
G
-
g(t,v2)\\oo
<
Cb(E)
x
exists
a
—
Cb(E)
is
constant L <
L\\vi -t>2||oo
Lipschitz
oo
such that
VtG[0,T], vuv2eCb(E).
^ReCb(E).
b-3 h: E
for
f c(s,Zl'z)ds
+
assume:
b-1 b
We
h(Z^z)
condition
boundary
—u(t,z)
for
=
weak solution to the
a
Theorem 4.2.3 Let E be
tion with
u(t,z)
assume
that, for
any
(t,z)
G
[0,T]
x
E, Zl'z
as
defined
m
(4.2.5) stays
m
E
[0,T].
there exists
(4.2.6)-(4.2.7)
m
a
unique solution
the weak
sense.
It
u
G
can
C/,([0,T]
x
be written
E)
which solves the
boundary problem
as
rT
u(t,z)
Proof:
Let
us
fix
some u
—w(t,z)
+
G
=
h(Z^z)
C/,([0,T]
x
E)
+
I
gZs' (s,us)ds.
and consider the PDE
b(t,z)—w(t,z)+gz(t,ut)
=
0
in
(0,T)
w(T,z)
=
h(z)
onE.
x
£,
(4.2.8)
(4.2.9)
SOLUTION FOR FIRST-ORDER
4.2.
Lemma
By
4.2.2,
EQUATIONS
47
know that
we
rT
w(t,z)
=
h(ZTz)
+
boundary problem (4.2.8)-(4.2.9) in the
and b-3, w belongs to Cb([0,T] x E).
solves the
tion b-2
Let
us
introduce the operator F
Cb([0,T]
:
E)
x
"'
g
(s,us)ds
weak
->
In
sense.
Cb([0,T]
x
E)
addition,
due to condi¬
with
r-T
(Fu)(t,z)
We
only have to prove that F is
by a similar argument as
shown
/ gz°'z(s,us)ds.
h(Z^z)+
=
a
contraction
on
in Section 4.1.
the space
Let us, for
C/,([0,T]
some
ß
x
G
E).
This
E+,
consider the
can
be
norm
111(11/3:=
which is
||«||oo-
to the supremum-norm
equivalent
e~ß(-T~^\u(t,z)\,
sup
(t,z)£[0,T]xE
Due to
b-2,
we
obtain for U\,u2
G
Cb([0,T]xE)
e-ß^\(FUl)(t,z)-(Fu2)(t,z)\
1
r
gZs' (s,u1>8)
j,
eß(T-t)
1
<
1
<
eß(T-t)
[0, T]
G
and
L\\ui-u2\\ß
—
\\Ui
G
z
-«2
a
contraction
there exists
a
on
unique fixed point
get that
a
partial
is also
further
c-1 b
assume
is
Fu2\\ß
L,
-\\ui
-
M2||/3,
(C/,([0,T] x E), \\ \\ß) with ß > L. Therefore,
C/,([0,T] x E), which solves the boundary problem
u
G
solution in the strong
us
<
sense.
assume
that E dR
assumed to have
sense
(4.2.6)-(4.2.7)),
differential equation
Theorem 4.2.4 Let
-
the normed space
in the weak
u
eß{T~s)ds
Thus,
E.
(4.2.6)-(4.2.7)
To
gZs' (s,u2>8) e-ß{T-s)eß(T-s)ds
||/3
\\Fui
and F is
f
-
(s,u2tS)Jds
L„
<
for all t
gZs' (s,u1>8)
eß{T-t)
gZs'
-
is
a
we
(i.e.
need
that all conditons
compact
as
well
a
classical solution to the interactive
some
of
further conditions:
Theorem
4-^-3
fulfilled.
Let
us
as
uniformly bounded,
continuous derivative
differentiable function v, gz(t,v) is differentiable
-i-v) for some suitable continuous function
gz(t, dz
g, fulfilling
c-2 For any
are
m
z
4-b(t,z).
with
-^gz(t,v)
=
CHAPTER 4.
48
there exist
•
INTERACTIVE PARTIAL DIFFERENTIAL
constants
some
K such that
L,
we
||£(s,t>s)||oo <L\\vs\\oo
for
•
any R >
0,
is
g
uniformly
continuous
EQUATIONS
may write
(4.2.10)
+ K.
on
[0,T]
x
M
E with M
x
{v
:=
G
Cb(E) | |M|oo<Ä}.
c-3 h G
Then,
Cl(E)
it
also the
is
Let
us
§^Zfs,z,
first discuss
entiating Zff
=
z
Gronwall's
Lemma,
bound denoted
Let
us now
by Lz-
discuss,
for
N into N.
have
are
=
Let
sup^Q/jy-)
uniformly
=
In
G
v
show
|H|oo
l +
by
Protter
with respect to z,
E),
defined
as
we
(1990),
Theorem V.39. Differ¬
get
fAiz^êf{u-z'f^
C/,([0,T]
is
uniformly bounded,
the
E),
x
e-^-\\^-zZY\\h'(z!rz)\ Jt ^Zf^ (s,vs)\ds)
<
+
<
e-^-Hz(Lh
<
^\\v\\ß
now
< oo.
x
+
directly can conclude that §^Zfs,z
analogy, let us denote Lh := ||^'||oo-
2LZL and N
us
C/,([0,T] x£)^ C/,([0,T]
we
e-*T-*\{Gv){t,z)\
Hence, for ß
:
and
JU(4*) f (^zZl>z)gz°z(s,vs)ds.
f* b(u, Z^z)du
+
to the
differentiable m the space variable
boundary problem (4.2.6)-(4.2.7).
is
which is well-defined
iz-*
By
=
E)
x
the operator G
(Gv)(t,z)
Let
C/,([0,T]
G
u
strong solution
analyze
us
continuous derivative.
bounded,
the weak solution
therefore,
Proof:
with
:=
{v
LzKT
a
+
x
+
K)e-KT-WT-*>ds)
LzLh.
E) \ \\v\\ß
compact operator
since g is
continuous, there exist
/VlNloo
Cb([0,T]
G
that G is
Further,
+
+
uniformly
(t\,Z\), (t2,z2)
<
on
2LZ(KT
N.
continuous
with t\
<
Since
as
t2 and
Lh)}, G maps
G(N) C N, we
+
well
\t\
—
as
t2\
ß-b
+
and h!
\z\
—
z2\
EQUATIONS
SOLUTION FOR FIRST-ORDER
4.2.
small
enough,
such that for
u
\u(t1,z1)-u(t2,z2)\
49
Gv
=
^r1 l~/r2\w(zfrx)\
\-^-Zt^Z2\\h'(ZtT1'zl) ti(Z%'Z2)\
<
-
+
-
\gzl^\\gz^
f \§-zZtsUZl j-zZl2'Z2\\9Zll'Z\s,vs)\ds
+
~
l
inequalities
are
()
r?t~\
_l
<
Lht + Lzt +
<
(Lh
+
uniformly
\t\
rytn
,z-\
t2\aLz
—
+ aTt +
LZ + aLzT + aT +
true
on
,zn
(s,vs)-gz* 2(s,vs)\ds
J \^-zZl2'Z2\\9Zs
+
The above
"frvJlds
LzTt
LzT)e.
Hence, by the Arzela-Ascoli theorem,
N.
G(N) is relatively compact. We can further show that
see this, let (vn) be a sequence in N with \\vn
0, i.e.
w||oo
Gvn. Then
uniformly on [0,T] x E to v. Set un
the set
G is continuous
N. To
the functions vn converge
—
—
on
=
\\u
—
UnWoo
\u(t, z)
max
=
0<t<T,zeE
rT
0<t<T,zeE
0
because of the uniform
to
9
max
=
—
continuity of
7t,z-
d~zZs
H
as n
un(t, z)\
—
)[gzl':'(s,vn>8)
-
gz''"(s,vs)
ds
oo
g and the uniform convergence of the functions vn
v.
We conclude that G is
fixed point
v
by
a
compact operator and therefore, G
:
N
N has at least
—
one
Schauder's Fixed Point Theorem. For
r-T
(Fu)(t,z)
the operator
on
C/,([0,T]
x
E)
=
,„
w.
d
,
=
Let
us now
,
,
+
h(Z1/)
of Theorem
differentiable in the space variable.
d
rt,z\
z\'z
I
/
gZs'
4.2.3, let
us
(s,us)ds,
assume
that
u
G
C/,([0,T]
x
E)
is
Hence,
,
f
+,,
d „,,\
/
d
yt.Z
(G-^u)(t,z).
consider the primitive with respect to
^-z(Fu)(t,z)
=
=
z
G
E of v, denoted
(Gv)(t,z)
v(t,z)
=
—u(t,z).
as u.
We may write
CHAPTER 4.
50
We therefore
INTERACTIVE PARTIAL DIFFERENTIAL
get that the function
u
may be written
u(t,z)
with function G
exists
that
u
a
[0,T]
—
E.
(Fu)(t,z)
On the other
unique fixed point of operator
is
solution
:
=
F in
hand,
+
as
C(t)
we
C/,([0, T]
x
uniquely defined. We therefore have shown
to the boundary problem (4.2.6)-(4.2.7).
EQUATIONS
know
by Theorem 4.2.3
E). Hence, choosing G
that there
that there exists
a
0,
get
unique classical
=
we
Chapter
5
Optimal Martingale
Measures under
Additive Processes
analyze the asset model of Section 3.4. As already noted, much attention has
given to the case where no contingent claim is present. The corresponding minimal
entropy martingale measure turned out to be of a particularly simple structure. Let us
now assume that the market participant wants to issue a contingent claim. This chapter is
dedicated to the analysis of the corresponding optimal martingale measure. We will show
that the optimal martingale measure can be identified via the solution of an Integro-PDE.
It frequently happens that options are written on underlying assets, for which no liquid
market exist, but where there is a liquid market in some closely related asset. Simply
using this proxy in hedging gives usually poor hedging performance (see e.g. Henderson
(2002)). This raises the question as to what are an appropriate price and the best hedging
strategy using only the tradable asset. We will see that this so-called two asset problem
leads to a PDE, which in essence is equal to the PDE of the one-asset problem. In fact,
we develop the theory in the two-asset world and we then give a short note with respect
to the one-asset problem.
The remainder of this chapter is organized as follows. First, we set up our model with an
additional nontraded asset. In Section 5.2, the main result, namely the partial differential
equation for the optimal martingale measure, is stated. The rest of the chapter is dedicated
Let
us
been
to prove this result.
5.1
Preliminaries
Suppose
we
processes,
have two assets whose discounted prices
more
=
Vi,tdt
+
(JhtdYtc
f are deterministic
standing assumption
a%
and
d(jt(x)
+
functions.
u(R)
This excludes
are
modelled
as
jump-diffusion
concretely, they satisfy
-w-^
where r\%,
Si, S2
Levy
processes with
*
(ßY
Further,
-
vy))
in this
,
section,
we
work with the
< oo.
jump parts of infinite activity.
51
We write
a
European
52
CHAPTER 5.
option
on
MARTINGALE MEASURES UNDER ADDITIVE PROCESSES
S2 whose payoff
asset
ever, cannot be
the
traded;
at exercise time T <
tradable asset is Si. We
only
about the fair price of the option
As
hedging strategy.
identification of the
Hence,
we
have
we
well
as
are
=
H(S2>t)-
Asset
how¬
S2,
confronted with the question
the related question about the
as
in Section
seen
optimal martingale
will concentrate
is B
oo
corresponding
directly relate to the
2.2, both questions
QB as
QB.
measure
representation (2.1.4).
well
as
its
processes
as
follows.
the identification of
on
Some technical assumptions will be needed:
Assumption
•
bounded and
are
r\l,al
from
Lipschitz
/i
is
and
uniformly bounded for
uniformly bounded from
If asset returns
modelled
are
Lipschitz
are
all t
G
[0,T]. f2
boundedness of
Assumption
/i
Remark 5.1.3
We
Levy
as
-
assume
H(s2)\
We consider any
are
payout H
excluded.
|—
seR>0
Let
H(s)\
dlogs
a
>>
}
H
us
locally
bounded away
from
—
1
are
fulfilled.
bounded.
bounded
is
as
all si,s2
well
as
(5.1.1)
eR>0-
Due to the boundedness assump¬
the
put-call parity,
show that put option
(5.1.1)
call option prices
payout patterns
are
is
fulfilled if
d
=
s\—H(s)\
ds
seR>0
sup
Put option with strike
f
-1
\
0
< oo.
K, H(s)
=
(K
—
s)+
is
not
m
for
for
all
se
all
s
G
(0,K),
(K, oo)
therefore
can
us
conclude that the Put option
introduce
X2>t
=
log(S2tt)
s\H'(s)\
Condition
is
(5.1.1)
simple
ensures
:=
that h is
to handle due to the
=
K.
payout fulfills condition (5.1.1).
well
as
h(X2>T)
X2>t
[0,T],
t G
needed for technical reasons, e.g.
for
Condition
sup
s+K
Let
m
however
[
We
function
However, by
prices.
d
payout pattern of
„//
and
functions
uniformly
are
rather restrictive.
is
C1(E>0).
G
sup
C1(E>0),
1,2,
(5.1.1).
the condition
the
that the
This assumption
expressed by put option
m
Obviously,
=
bounded away
is
the first two conditions
processes,
LH\logsi-logs2\
<
tion, European call options
be
a%
i
above.
that the price process Si is
ensures
5.1.2
\H(si)
included
[0,T],
continuous
is
The boundedness conditions of the third condition
might
on
For
zero.
—
•
functions
continuous
/j : [0,T] x supp(z/)
(—l,oo)
uniformly for any x G su.pp(u).
•
of price
We restrict the class
5.1.1
as
the function h
H(S2>T)
Lipschitz
=
:
E
—
E, which
is defined
as
H(S2fiex^T).
continuous
following property:
(with Lipschitz
constant
Lh)-
MOTIVATION AND MAIN THEOREM
5.2.
Lemma 5.1.4
Given
dX2>t
then, X2>t
=
53
r]2ttdt
=
\og(£(X2> )t)
+
a2ttdYtc
may be written
dX%t
fj2ttdt
=
+
+
dyf2(x)
+
d(f2(x)
*
(ßY
"y))
-
,
as
d%tdYt
(/iY
*
"y))
-
with
v2,t
=
m,t
=
f2,t(x)
A
proof
=
of Lemma 5.1.4
us
discuss the
sponding setting,
Am
Let
^l,t + f (^og(l
log(l + f2,t(x)),
+
m,t-
can
f2tt(x))
-
be found in Goll and Kallsen
f2tt(x)y(dx),
(5.1.2)
(2000).
Motivation and Main Theorem
5.2
Let
C2,t,
us
=
Optimal Martingale
we
Sltt-(alt+Yflt(xMdx)'
introduce
Equation of Section
Measure
3.3.
In the
corre¬
have
Xi>t
:=
S^t-A^
°*
well
as
as
=
(pt
WtM{x)
Sl'*-<71'"
Siyt-(j)f By
:=
=
Sl't~fl't{x)-
equation (3.3.3),
we
may
write
aB +
(-(of
cB+
[\VtL(x)
-
-
\i,to-i,t)2
$tM,to-lt)
+
($t + \i,t)fi,t{x)
+
dt
$Âi,tfi,t(x)y(dx)dt
'of -Q>t + \t)oi^dYfc
+
((log(l
We further know from
-
(3.3.2)
that
o-iX
Besides
even
that,
we
do not know
know whether these
Xifi(x)
+
of
WL(x))
and
WtL(x)
-
$fi(x)) ßY)T.
are
Section 3.4 does not work here since
must be defined in such
B
=
deterministic
=
or
a
way that
(5.2.2)
0.
about the structure of
we are
(5.2.1)
*
/ fi,t(xWtL(x)v(dx)
more
objects
+
WL, aL
stochastic.
The
(p. We do not
argumentation of
or
confronted with the additional stochastic term
h(X2>T).
(5.2.3)
CHAPTER 5.
54
However,
order 0
since the
=:
MARTINGALE MEASURES UNDER ADDITIVE PROCESSES
Levy
To < Ti <
measure
such that
...,
is
finite, jump
we
times of Y may be counted in
increasing
may write the second term of the RHS of
(5.2.1)
in
the form
'
'log fl
Xifi(x)
-
Y, [log
(l
+
WL(X))
$fi(x))
-
Ai>Tl/i>Tl(AyTl)
-
+
*
ßy
WB(AYTj)
-
$rJi,Tt(&YTt)
n<T]-
i=i
The LHS of
jumps
(5.2.1)
of X2.
is
Introducing
f2,t(x)) -u(t,y),
+
therefore work with the Ansatz that there exists
log
(l
\i,tfi,t(x)
-
to the
the notation
&ly(x) :=u(t,y
we
(5.2.3), directly exposed
random variable which is, due to
a
+
WtL{xj)
function
a
$tfltt{x)
-
=
u
such that
(5-2.4)
A?Ai(»,
jumps on the RHS of (5.2.1) correspond to the jumps of some function u along
the paths of process X2. In addition, we set u(T,*)
ah(*) in E and we assume that u
i.e.
the
=
is
sufficiently smooth such that Itô's Lemma
are no jumps in (rt, rl+i), we get
may be
applied. Taking
into account that
there
u(rl+i,X2>Tr+1_)
=
/
u(Tt,X2>Tr_
-
/2,T,(AyTi))
+
du(t,X2}t)
'(n,Tz+1)
d_
dt
'(n,TI+1)
u(t,X2tt-)
+
Ctu(X2tt-)
dt
d
f
/
'(r„r,+1)
au^-u(t,XU-)dYfc
9y
with
£tu(y)
(r\2,t-
:=
We may therefore rewrite
cB
+
f2,t(x)is(dx))—u(t,y)
equation (5.2.1)
+
àhd^ ;u(t,y).
2
dy
(5.2.5)
as
u(0,0)
T
1
^
-(of
-
°~t
ensure
~
(0*
d
^^
Xi,to-i,t)2
+
+
Ai,t)<Ti)t
that the RHS of
~
<PtKtoX)t + ju(t,x2)t-)
/ (wtL(x)
+
We must
j
d
-
-
($t
+
A1)t)/1)t(x)
G2tt-pr-u{t,X2tt-)
dy
(5.2.6)
is
a
+
+
Ctu(x2>t_)
$Ai,tftt(x))is(dx)
dt
(5.2.6)
dYtc.
constant. A solution to this
problem might
be to require
^(aL
+
-
\io-i)2
+
$Xiaf
/ [W L(x)-(<Pt
+
+
—u(;X2>_)
\i)fi(x)
+
+
Cu(X2>_)
<p\ifi(x) )u(dx)
=
0
(5.2.7)
5.2.
MOTIVATION AND MAIN THEOREM
with
u(T,~k)
ah(~k)
=
in combination with
oL
By equation (5.2.8)
which, replaced
WtL(x)
,
in
y
:=
Let
X2>t-.
WtL(x)
get
we
f fit(x)Wf(x)u(dx)
J
J J1'ty
,
V
I
-
;
V
^
;
-
a{t
5.2.9
AM,
to
-ä2,tdu(t,X2t.)
,
f fi,t(z)WtL(z)u(dz)
Ai
a\t
dy
-^M
(5-2.10)
functions
are
depending
onti(
u(t,
:=
)
E
:
E and
—
introduce
:=
\(°ï-\to-i^2+ itXA,t
+
(5.2.7)
(5.2.2),
Am/mW-
and
us
fcy(*,Mt)
(x)-fi,t(x)
(5.2.8)
dy
equation (5.2.4), leads
A»,
(pt, of
52^-u(.,X%_).
=
°~i,t oy
-1 +
Note that
d+\i)oi
&2t d
-^Fnt,V
=
exp
=
-
in combination with condition
t
(pt
55
(5-2.11)
J(wtL(x) (J* AM)/M(x) JtA1)t/12)t(x))I/(dx).
+
-
has then the form of
+
Integro-PDE for u, whereas u has also to fulfill the
boundary condition u(T,*)
ah(*). In the following, we will show existence of such a
function u and the appropriateness of the chosen approach. To formalize these results,
an
=
we
provide
the solution in form of
Theorem 5.2.1
1.
Let
us
(Main
a
Theorem
theorem:
I) Suppose
ju(t,y) + Ctu(y) + ky(t,ut)
u(T,y)
with the second-order operator
exists
may be written
5.1.1 and 5.1.2 hold.
a
=
0
for
all
=
ah(y)
for
all
(t,y)
u
m
[0,T)
x
E, (5.2.12)
(5.2.13)
y G E.
£t and the function k defined
unique solution
G
m
(5.2.5)
Cb ([0,T) xl)fl C1,2((0,T)
and
x
(5.2.11),
E),
which
as
û(t,y)
with
Assumptions
Cauchy problem
consider the
respectively.
Then, there
that the
=
E
ah(X^y)+ I
k*'(s,u8)ds
(5.2.14)
Xl'y defined by
Xls'y
=
y+ I
t
fj2,udu+ I
Jt
gvJY:-
(5.2.15)
CHAPTER 5.
56
2.
Having
(p
with
MARTINGALE MEASURES UNDER ADDITIVE PROCESSES
introduced
u
WL
above,
defined
Q, defined by
and
measure
as
we
fix
m
equation
u
dQ
cB
=
—«(0,0) equals
introduce the
/
^
strategy
(5.2.10).
and
rT
'
the
us
(5.2.9)
jc^
exp
dP
with
Let
u.
=
(j)BdSi}t
Then,
(pB
the
=
i^—(pt
probability
aB\,
+
jo
optimal martingale
measure.
Remark 5.2.2
1.
With
auxiliary function
Taking
u
=
u
WtLy(ut,x)
one
space
=
describes the
WBy(ut)(x)
parameter
We will later show
[0, T]
x
E
x
supp(z/).
a
function
will
we
that,
drop
since u
is
of
price
some
into account that
treat the latter
Example
arguments:
su.pp(u),
m
of
the
and the
parameter,
auxiliary function
to
arguments
time
one
easen
uniformly bounded, WL is uniformly bounded on
aL and (p are uniformly bounded on [0, T] x E.
the
contingent claim
by Corollary 2.2.3,
can,
cB
5.2.3
(The
a
—«(0,0),
=
case
of
the above
problem
for the
following example,
has to be solved
contingent claim. In the
no
case.
Case)
MEMM
Let
with the Ansatz that the solution u* has the
be reasonable
u*(T,y)
since
Under this assumption, let
us
consider the
B
case
H(S2>T)
=
=
us
=
property
fj-u*(t,y)
=
0
on
(0,T)
x
wL,
*
,
Wtty{ut ,x)
E.
and
mar¬
start
This
0.
discuss
m
detail the
function
k
m
the PDE
(5.2.12). First,
^(f
iff
fht(x)
^
I
exp
=
hÀ^WtLM,z)u(dz)
-
g
+
T
u
Xht) j
-1+WmOz)-
Due to
We
0.
(5.2.10) provides
equation
ai
be
-(c°-cB).
=
from Section 3.4 that this problem corresponds to finding the minimal entropy
tingale measure. Let us investigate the solution of the PDE (5.2.12). We propose to
We will
ut.
the notation.
know
might
pair
as
contingent claim B and the
we
(Zt)te[o,T\-
=
(Zt)te[o,T]-
=
with many
7rs(B;a)
Taking
process Z
density
We conclude that
4- The utility indifference
written
is
process Z
density
R and another
m
Whenever appropriate,
3.
may describe the entire
(p, equation (5.2.8) defines aL. Hence, by Remark 3.2.3, the
and
(aL,WL) uniquely
2.
we
u,
see m
Xt
are
(5.2.9)
Section 5.3 that equation
independent of ul
and
(5.2.8),
2
(pt
we
(5.2.16)
(5.2.16) defines
unique
that also
WL
f fi,t(x)WtL(x)v(dx)
î
and y,
we
see
is
function WL.
Since
independent of ul
fi,
and y.
may write
=
5
a
°f
a
=
(<f>t
+
i,t
AM)crM.
Ai)t,
(5.2.17)
MOTIVATION AND MAIN THEOREM
5.2.
Hence,
is
also
(pt
and
of
independent of u* andy.
are
-ß-u*(t,y)
constant; hence, the Ansatz
a
l,From equality (5.2.16)
as
well
=
-ai,t$t
\
~
h,t(x)
0
=
(5.2.17),
as
fi,t(x){Wt(x) + 1
=
57
for
t
fixed, ky(t,u1)
justified.
is
we
We conclude that
get
Xi,tfi,t(x))iy(dx)
-
\to-\t + /
fi,t(x)v(dx)
-
AM / f2t(x)u(dx)
\ 4>tf\,t(x) \v(dx).
exP
Since
Vi,t
Î
_
M,t
(pt
must
the condition
S
us
=
Si
now
=
discuss the
S2,
which
case
~
To work in the
single
rftdt
+
we
\9t-
on
the tradable asset
+
only
dyf(x)
*
(/iY
-
"y))
have to make the
à2 d2
ß
x
„
adjustments
+
-±-—u(t,y)
as
(compare
with Lemma
ft(x)
=
u(y
=
5.2.4
Merton's jump
ft(x))
Let
(The Single-Asset Case)
diffusion
Brownian motion and
a
-
u(t,y),
log(l+ ft(x)),
Cauchy problem (5.2.12)-(5.2.13).
simplified asset model:
Example
+
5.1.4)
in the
a
is defined
j ft(x)v(dx))—u(t,y)
Aly(x)
with
contingent claim
otdYtc
r
/
=
0,
as
asset case,
£tu(y)
=
Section 3.4-
m
where the
define
we
5Vl
well
f ftt(x)v(dxY
/i,t(a;)(exp \$tfi,t(x)} ^)v(dx)
already identified
dSt
as
°lt
+
fu
ai,t$t + Vi,t + /
Let
—
model where
we
us
analyze
Let
assume
Poisson process. In
our
us
the
single
consider
a
St-
=
r]dt
+
odYf +
setting,
dip* (/iY
V
very
case
simple
that the asset process
is
at hand of
version
driven
the asset process has the
ics
—^
asset
-
vy) )
/t
of
by a
dynam¬
CHAPTER 5.
58
for
a
constant p >
—
contingent claim B
we
is
MARTINGALE MEASURES UNDER ADDITIVE PROCESSES
Let
1.
us
defined
assume
that
St- As
the asset
on
v(R)
X.
=
In
one can
addition,
directly
we
that the
assume
by equation (5.2.9),
see,
write
can
Pwïv(p)x
d
2
9t
u
^
—K-u{t,y)
=
a2
dy
WtLy(p)
whereas
Wt(p)
=
must
exp
fulfill
V
2——2T'
a2 + p2X
the equation
\u(t,y+ log(l
+
d
p))-u(t,y)-p
„
,
,
—u(t,y)
dy
WtLy(p)pX
a2
a2 +
p2X
W
.1
a2 +
p2X'
simplified version of a single asset problem reduces the com¬
problem only gradually. The mam drawback, namely the non-lmeanty of
plexity of
Wfy(p) with respect to the function u, stays. Therefore, we conclude that the mam diffi¬
culty m pricing contingent claims via a utility-indifference argument lies within the jumps
m the underlying price process.
the reduction to this
Hence,
the
In this
Section,
To make this
measure.
As
a
introduced
we
intuitive
proof,
we
for
identifying the optimal martingale
proceed as follows:
approach
approach rigorous,
to the
preparation
an
we
shall
discuss in Section 5.3 the function
that the function is well-defined and present
some
C1,2([0,T)
properties of it.
We
ensure
5.4,
we
Cauchy problem (5.2.12)(5.2.13) with u G Cb ([0,T) x E). The proof will be completed in Section 5.5, where,
performing the verification procedure of Chapter 2.4, we prove that the resulting strategy
(pB defines the optimal martingale measure QB.
show there exists
a
classical solution
G
u
implicit representation
Having only
an
defined,
i.e.
we
Wfy(ut)
solving (5.2.10).
show that each
u
Before
G
of
Wfy,
we
Cb ([0,T)
let
doing this,
to the
WL
Discussion of Function
5.3
x
E)
Wfy.
In Section
x
have to
ensure
E) uniquely
us
start with
that the function is well-
defines
some
a
bounded function
auxiliary
results:
Lemma 5.3.1
1.
l°°(su.pp(v)), the
function on su.pp(u),
being
E, given as
ifk ' supp(z/)
Let
ß
k
>
0, /
G
a
set
of
which
bounded
is
functions from supp(z/)
from above. Then,
bounded
—
ipk(x)
is
2.
well-defined
Let
us
=
exp
lk(x)
-
ßf(x) / f(z)ipk(z)u(dz)
and bounded.
define
$fc
:=
/
f(x)ifk(x)u(dx).
into
the
R,
and
function
DISCUSSION OF FUNCTION WL
5.3.
//
functions ki,k2
have two
we
ki(x)
ki(x)
then
get ^kl
we
^
>
59
l°°(su.pp(v))
G
k2(x)
k2(x)
<
>
with
x
G
V
x
G
su.pp(u), f(x)
supp(z/), /(x)
V
<
0
>
0
(5.3.1)
•
Proof:
ad 1. Let
us
start with
$
Let
us
=
show that there exists
purpose, let
us
the equation
considering
/
f(x)
a
unique value $fc
exp
j k(x)
ßf(x)$ >u(dx).
-
G
E which fulfills this
equation. For this
define
H(z)
z
=
f(x)
—
exp
<
k(x)
—
ßf(x)z
> v
(dx).
Since
lim
f(x) exp{—ßf(x)z\
—
J
K
l
'
f
l
J
K
'
J
=
we
is
get lim^oo H(z)
continuously
=
oo
and,
=
>
there exists
a
us assume
„
,
'
H(z)
for symmetry reasons, lim^-oo
<
that §k > 0.
$fc
/ ßf2(x) exp ik(x)
1 +
=
—oo.
Further,
H
G
Hence,
E such that
H(§k)
=
0. We further
\expk(x)\ / \f(x)\v(dx)
max
icGsupp(^)
we
:
get
f(x)exp\k(x)-ßf(x)$k\v(dx)
=
<
ßf(x)z\u(dx)
-
0.
unique $^
|$fc|
Let
£) [~
f(x) < 0
OO
differentiable with
ft-H(z)
Therefore,
{
{
z^oo
/
f(x)exp\k(x)
/
f(x)exp<k(x)>u(dx)
-
ßf(x)$k\v(dx)
J{f(x)>0}
<
l
J{f(x)>0}
<
ma*
{expA.(x)| /
zGsuppM I
<
max
<
icGsupp(^) I
The lower bound
can
be shown in
J
J
J{f(;c)>oi
expfc(x)> /
) J
exactly
the
/(*)„(&)
|/(x)|z/(dx).
same
way.
can
show that
(5.3.2)
CHAPTER 5.
60
Let
us now
MARTINGALE MEASURES UNDER ADDITIVE PROCESSES
define the bounded function
<p(x)
:=
{k(x)
exp
-
ßf(x)$k
Obviously,
/
f(x)(p(x)v(dx)
/ f(x) exp \ k(x)
=
Hence,
therefore,
ad 2. Let
us
we
{k(x)
exp
=
conclude that tpk
that
assume
§kl
<
Vk2(x)
f(x) I f(z)p(z)v(dz)
ip is well-defined and bounded.
exp\k2(x)
directly get
we
for any
kx(x) -ßf(x)(®k2
-
l
J
\
>
1
V
x
G
<
1
V
x
G
supp(z/), /(x)
supp(z/), /(x)
-
G
x
supp(z/)
that
$fcl)
<
0
>
0
hence,
tpk2(x)
However, this leads
Therefore,
us now
Corollary
defines
a
we
to
show that
-
therefore
also
(p
Proof:
For any
$fci
§kl
Wf,
Under
function
aL
(t, y)
<Pkiv{x)
V
0
G
x
yxe
<Q
are
G
=
>
/ f(x)(tpk2(x)
supp(z/), /(x)
supp(zy); j^
<
0
>
q
<pkl(x))v(dx)
>
0.
•
is well-defined:
G
5.1.1
l°°(su.pp(v))
continuous and
[0,T]
=
-
$fc2.
Assumption
Wfy(ut)
and
,
contradiction since
a
must have
5.3.2
>
ipkl(x)
$fc2
Let
:=
-
$fc2. Then,
VkAx)
and
ßf(x)$k \v(dx)
get that
we
(p(x)
and
-
x
E,
we
as
Cb ([0,T) x E), ut uniquely
fulfills equation (5.2.10). WL(u) and
well
which
as
uniformly
may write
explkly(x)-ßtf(x)
u
G
bounded
for
all
equation (5.2.10)
(t,y)
G
[0,T)
in the form
ft(z)pk*Jz)u(dz)
x
E.
DISCUSSION OF FUNCTION W
5.3.
61
with
VK,y(x)
:=
ßt
=
WtLy(ut,x)
1,
+
—2~,
a
ft(x)
Xfht(x)
-
i,t
fi,t(x),
:=
and
H,y\X)
\,y\X)
'=
f
f
-fl,t(X)
Since ut
Cl(R),
G
\
have
we
Ai,t V 1
L
k*y
which is
gets by
the
2
(t,y). Hence,
Wfy
The function
restricts the space
we
introduce
a
and
|/gR
we
with
the
(see
G
as
-§-u(t,y)
are
uniformly
Cb ([0,T) x E). On the other hand,
(1986), Section 4.7) that
one
WtLy
is continuous in
bounded.
ensured.
(t,y).
However,
if
one
For this purpose,
us
fix
(1,2].
E
some r
Q,
We recall the
definition of the
space
Cb'r
norm
L
>
07
equip with the
the definition of
Corollary
apply
Zeidler
conclude that
we can
specify
we
[w
:=
*
W^ II °°*
of functions C®,L
Clbr(R) | II^IU
G
<
Q, \w'(y)\
as
follows:
< L
norm
C®'L,
5.3.4 Let
11^11°°
•—
the subset
\\w\\y
By
may
suitable subset of functions:
CfL
which
we
/ fi,t(x)fkiy(x)u(dx)
=
lll^lll
For
m
well
as
—
Notation 5.3.3 Let
Cbi(R)DCr(R)
A^y
E since
x
l°°(su.pp(v)) is not uniformly
C^(E)
C^(E) appropriately, boundedness can be
:
5.1.1 and
lj->y
$fc?,y
is continuous in
J
ahtdy
alt
t,y
[0,T]
Function Theorem
Implicit
-i
ä2,t d
TTu{t,y)
+
and therefore also Wf, is well-defined and bounded.
direct consequence of
a
2
l°°(su.pp(v)) by Assumption
'
on
ffiÄz)v(dz)
S
alt
pk*
'
WL is uniformly bounded
+
G
above Lemma 5.3.1.1. Hence,
bounded,
IfUz)u(dz)\
(im
\\
we
(t,y)
G
'=
IML
+
\w'(y)\-
get
[0,T]
x
E be
fixed. Then,
WtLy(v)
is
uniformly
bounded
for
allveCyl'L.
Even the
following
statement
Lemma 5.3.5 For
(t,y)
G
can
[0,T]
be made for
x
E
fixed,
Ky C?'L
Lipschitz continuous, uniformly
independent of y G E.
is
Wfy:
-
HsuppM)
with respect to t
G
[0,T],
and with
a
Lipschitz
constant
CHAPTER 5.
62
Proof:
Hence,
Due to
(t,y) fixed, WL is
Lipschitz continuity of WL, i.e. we
Corollary 5.3.4,
know that for
we
only has to show local
0 fixed, there exists a Lipschitz
one
for any
MARTINGALE MEASURES UNDER ADDITIVE PROCESSES
c >
constant
bounded
on
C®'L.
have to show that
Lc such that
l^>i)-^f>2)||oo<£cK-^|
for all Vi,v2
G
Cff,L
with
\vi -v2\L
In the
following,
consider v0 +
for any v0
qh
11/11100
and q
G
[0,1].
C®'L,
h
Cb'r(R)
G
with
We will show that
5.3.1,
we
showed that for
ipk{x)
exp
:=
$k
are
c.
\h'(y)\<c
0,
=
\\WtLy(v0 + qh)
In Lemma
G
<
ß
0 and
>
<^ k(x)
WtLy(v0)\\oo
-
-
f,k
G
<
Lcq.
l°°(su.pp(v)),
ßf(x) / f{z)<pk{z)v{dz) \,
f(x)ipk(x)u(dx),
-
well-defined.
We know from
Corollary
5.3.2 that
Vk*{q)
=
may write
we
WB(vo
+
qh) -Xifi
(5.3.3)
+ l
with
ß
2
a
'
f(x)
=
fi(x),
k*(q,x)
:=
kl(x)
+
q^(x)-fi(x)^h'(y)),
and
kl(x)
:=
A;°(x)-f\(x)
^v>0(y)+rj
-Ul
where
~_Wi
V
Due to
(5.3.3),
we
:
Xi
(1
+
ffi(z)u(dz)
ffî(z)Hdz)\
)
-5
have to show that for all q
ll^fc*^)
—
G
-5
[0,1]
fk*(o)\\oo
_
Lcq.
.
(5.3.4)
5.3.
Let
DISCUSSION OF FUNCTION W
us
therefore
63
analyze
\<Pk*(q){x) -^k*{o)(x)\
k\ (x)
exp <
=
\qAhy(x)
exp
-
y
l
fl(x)q^h'(y)
-
Gl
$fc*(g)^
*fc*(o)—:2—
exp
Gl
fi(x)
exp{fcî(x)-$fc*(0)^j
=
(J-l
v
Since wo and
x
su.pp(u).
G supp(z/)
G
xo
fx0(q)
We have
v'0(y)
and
=
us
us
AM
(5.3.5)
^2
uniformly
bounded for all
For this purpose, let
investigation.
-
/i(^o)
$fc* (o)
-
the upper and lower bounds of
us assess
us
fix
an
fxo
for g
1.
G
[0,1].
introduce
k+(q,x)
:=
k~(q, x)
:=k\(x)
k\(x)
supa;esupp(l/) |/i(x)|
-
$fc-(0)
.
$k*(q)
<
-
+
-
gc^sign/i (x)
qcLs\grvf1(x)
We know from Lemma 5.3.1.2 that
$fc*(0)
< $
k+(q)
-
(5.3.6)
$fc+(0)-
consider the upper bound
$fc+(c)
The existence of the derivative
tion Theorem for Banach
$
We
the RHS is
on
-
$fc-(?)
Let
the first term
$fc.fc*(0)
v
|ç(aJ(x0) fi(x0)yti(y)^j ($fc.(g)
0. Let
2
(<$>k*{q)
-
0~l
the term
analyze
we
exp
=
1 +
c
fi(x)q—h'(y)
The second term needs further
fxo(0)
:=
-
bounded,
are
For this purpose, let
with cL
(72-
\ qAh(x)
exp
s
get, due
to
<pk+(q)(x)
d
dq
can
=
exp
exp <
®k+(q)
=
d_
$fc+(0)
be
Spaces (see
fi(x)
k+(q)
-
ds
$k+(s)ds.
guaranteed by
e.g. Zeidler
<j k\(x)
k+(q,x)
+
qcLsignfi(x)
AW
exp
application
(1986),
$A
äf^k+{q)
\k+(q,x)
Section
-
4.7)
d
fi (x)
$
$
CÏ
k+(q)
k+(q)
-
a2
Implicit
to the
(,
o\ dq
—
of the
u(dx),
Func¬
equation
^~ \v(dx).
fx(x)
fi(x)
/ fi(x) cLsignfi(x)
x
an
CHAPTER 5.
64
such that
MARTINGALE MEASURES UNDER ADDITIVE PROCESSES
(recall
may write
we
<
Since
by
k+(s)
some
G
/°°(supp(z/)),
we
Applying
the
^k-(g)
Taking
into account the
exp{qcLK}
1 >
-
^P-^k+(q)
know from Lemma 5.3.1.1 that <~pk+(q) is
G
s
[0,q\. Therefore,
$fc+(o)
-
~
—
H/f^W*
to the lower
steps
same
k+(q,x)
<
exp
/ |/i(x)|(^fc+(ç)(x)z/(dx).
cL
constant K* for any
$fc+(g)
{
CL
=
=
i f h (x) (sign/i (x)J ^fc+(g) (x)u(dx)
d
V
<pk+(q)(x)
that
bound,
inequalities
sup
icGsuppM
fx(q)
(5.3.6),
>
fXo(q)
bounded
get
get
\fi(x)\u(dx).
~cLqK*
>
of
we
uniformly
\fi(x)\u(dx).
cLqK*
<
$fc-(o)
we
we
get the following bounds:
inf
>
icGsupp^)
fx(q)
>
exp{-qcLK}
-
1
with
K:=l +
X*SUP^supp(;)l/l(:r)l i/ |/i(x)|K^).
^î
We therefore have for q
G
[0,1]
that
sup
icGsupp(^)
and the
Let
Lipschitz continuity
us now
fix
a
function
Lemma 5.3.6 For
(t,v)
v.
G
|/x(g)|
<
q(exp{cLK}
of WL with respect to
We
[0,T]
v
G
1)
-
C®'L is shown.
get the following result for
C61,r(E)
x
Wf+(v)
:
E
—
/°°(supp(z/)):
/bed,
H/^(W):E^/°°(suppH)
zs
Holder continuous
coefficient
Proof:
is
ß
=
r
The claim
—
m
y
uniformly
with respect to
(t,y)
m
compact subsets. The Holder
1.
can
be shown in
for any compact set K C E
fixed,
a
similar way
there exists
a
as
Lemma 5.3.5. We have to show that
constant
Hk
as
well
such that
\\K^)-wL^)U<HK\yi-V2\ß
for any yi,y2
G
K. Since
WL is bounded,
lim
one
only
has to check that
WByi(v)-wBy2(v)\u
as
some
ß
G
(0,1]
DISCUSSION OF FUNCTION W
5.3.
is bounded
on
well
E, \h\
h
as
G
K for
suitable choice of
a
1, such that
<
65
yo +
qh
In the
ß.
following,
[0,1].
K for all g G
E
choose
we
Let
yo G K
some
as
introduce
us
qh
k*(q,x)
v'(y0
kl(x)+
:=
-fi(x)
—
[v'(yo
+
f2(x)
+
qh)
+
z)
v'(y0
-
+
z)
dz
v'(y0)],
-
Ol
k\(x)
Avyo(x)-fi(x) -v'(y0)
:=
+
ff
-Ol
^defined by (5.3.4).
and
\Vk*{q)(x)
We
get
fk*{o){x)\
~
<^ k\(x)
exp
=
$fc*(o)
-
fi(x)
\Uq)\
o\
with
rqh
fx(q)
/
Jo
exp
:=
[v'(y0
+
f2(x)
z)- v'(y0
+
z)]dz
+
,a2r
.„
f^^v^yo +
en
-
qh)
-
v'(y0)]
(^(,)-^(0))^}-l.
J
°i
Let
fix
us
Since
v
G
Cb'r(R),
(dependent
su.pp(u).
xo G
an
on
Let
us
qh)
+
fXo
the upper and lower bounds of
assess
\v'(yo
have
we
K).
We
v'(yo)\
—
HvK\qh\r~l
<
for
a
for q
G
[0,1].
suitable constant
HVK
introduce
k+(q,x)
:=
k*1(x)
k~(q,x)
:=
kl(x)
+
-
qr~1cLsignfi(x),
qr~lcLsigiifi(x),
with
cL:=\h\r-l\2\\v'\\oc+
By
the
same
constant
Q*
reasoning
of Lemma 5.3.5,
proof
.
get that there exists
we
such that
-qr~lcLQ*
Hence,
in the
as
\fi(x)\^HvK
sup
we can
<
$fc-(g)
-
$fc-(o)
<
$fc*(g)
-
$fc.(0)
<
$fc+(g)
-
$fc+(0)
<
qr-lcLQ*.
conclude
exp
<j
,r-l
gr
QcL
[>
-
1 >
sup
icGsupp(^)
fx(q)
>
fXo(q)
inf/*(<?)>
>
exp
icGsupp(^)
^ -q^QcL\-l
with
Q:=l
+
Q
,
SUPlEeSUpp(» l/li^jl
o\
and therefore
sup
icGsupp(^)
We therefore conclude that
|/x(g)|
Wf (v)
is
<
qr
locally
l(e^p{cLQ}
-
1).
Holder continuous with
ß
=
r
—
1.
some
CHAPTER 5.
66
MARTINGALE MEASURES UNDER ADDITIVE PROCESSES
Existence of Classical Solution
5.4
In this
section,
we
will show how Theorem 4.1.6
(5.2.12)-(5.2.13)
E) n C1>2((0,T) x E)
be
can
applied
such that the existence and uniqueness of
Let
us
review the
a
to the
solution
Cauchy problem
u
G
Cb ([0,T)
x
is proven.
partial
differential equation
d
dt
u(t,y)
+
Ctu(y)
+
ky(t,ut)
=
0
with
ky(t,ut)
=
-{af(y)-\t(Ji^
+
+$t{y)Xi,t<rlt
/ (wtLv(x) ($t(y) Xi,t)fi,t(x) Hy)\tftt(x))v(dx)
+
-
+
and
rt{y)
=
fWtLy(x)fht(x)v(dx)
-
(Tit
My)
This PDE
can
=
be written
0~2t
du(t,y)
vi,t
dy
f fi,t(x)WtLy(x)u(dx)
-
a
Ai,t.
i,t
as
d_ u(t,y)
dt
+
qu(y)+gy(t,ut)
=
0
with
d
C*tu(x)
:=
ßt
=
ßt—u(t,y)+
fj2tt
-
ah
u21
d2
^
2
2u(t,y),
Q~2,t
/ f2,t(x)v(dx)
Vi,t
-
VU
/ fi,t(x)v(dx)
and
f fi,t(xWtLy(x)u(dx)^
gv(t,ut)
Al.tCTl;
au
/ fi,t(x)
+
1
^
-
Xittfi,t(x) u(dx)
+
Remark 5.4.1
a
Lipschitz
follows:
continuous
We introduce
function
-
g
function. The way
auxiliary function
an
/ fi,t(x)WtLy(x)u(dx)
i,t
/ WtLy(x)u(dx)
The continuous
L
ö
a
X2
:
/ flt(x)u(dx).
[0,T]
we
x
C
l,r
(5.4.1)
Cb(R)
is
circumvent this technical
g, which
fulfills
general not
problem is as
m
the conditions a-2 and a-3
EXISTENCE OF CLASSICAL SOLUTION
5.4.
of Theorem 4-1-6. We will
partial differential equation
67
show that the solution
ju(t,y) + etu(y) + ~gy(t,ut)
u(T,y)
fulfills
We then conclude that
Let
Step
Let
proceed
in three
=
consider
Due to the
bounded
G
slope
(1,2]
a
in
and
way that
a
for
all
=
0,
(5.4.2)
=
ah(y)
(5.4.3)
(t, y)
function g
auxiliary
[0, T]
G
E.
x
partial differential equation (5.2.12)
[0,T]
:
x
Clbr (R)
->
Cb(R)
function of the form
of
quasilinearity
structed in such
r
to the interactive
steps:
9(t,v)(y)
Let
E)
(5.2.13).
auxiliary
an
gy(t, ut)
also the solution to the
is
Definition of the
1:
us
u
condition
boundary
us
x
the equation
~gy(t, ut)
with
Cb([0,T)
G
u
=
g(t,K(v,t,y))(y).
f(WL(v))(x)fi(x)u(dx)
function
a
will be transformed to
v
§-v(y),
in A^ and
k
will be
con¬
bounded function with
a
of y.
neighborhood
L, C, Kh be positive constants, then
K(v,t,y)
define
we
K2oKi(v,t,y)
:=
K2(Ki(v,y),t)
=
and
•
Ki
Cb'r(R)
:
xR^
Cb'r(R) capping
neighborhood {i£R | \x
the
II/2lloo
Ki
sign(v'(y))L
as
well
if
|t/(y)|
addition,
re2
:
Cl'r(R)
maximum
k2 is
if
v
is
a
11 /211 oo}
sup
(i,ic)e[0,T]xsupp(»
Cb'r(R)
G
v
of y
way that
G
to
a
maximum of L in
E, whereas
1/2,4(2;)I
< 00.
=
=
1—
independent
<
of y
on
the set
[0,T]
Cl'r(R) capping
^ + (T
t)C.
x
-»•
{x
G
| \y
E
—
x\
the absolute value of
<
v
11/21100}- (5.4.4)
G
C61,r(E)
to
a
—
constructed in such
H«,
<
of
as
(ki(v, y))'(x)
•
=
for
:=
y\
slope
v if H^'Hoo < L,
Ki(v,y)
(ki(v,y))'(y)
L as well ast;^ Ki(v,y) is Lipschitz continuous, uniformly
G Cb'r(R) fixed, y
Ki(v,y) is locally Lipschitz continuous
is constructed in such
in y. In
—
the
KH +
a
way that there exists
(T-t)C-e, K(v,t)(y)
=
some
e
>
sign(v(y))(KH
K2(v,t)
t)C) if \v(y)\
0 such that
+
(T
-
=
v
=
CHAPTER 5.
68
KH
+
(T
ß'\v'(y)\.
with
—
MARTINGALE MEASURES UNDER ADDITIVE PROCESSES
t)C,
In
norm
as
well
for y
addition,
||
\\y.
•
there exists
as
let
fixed,
function, Lipschitz
Lipschitz constant L2(l), independent
Lemma 5.4.2
Having
1.
2.
Cb'r(R) \ \w'(y)\
E
a
v
E
C*'1
:
x
C*'1, uniformly
[0,T]
—
<
<
1}
C*'1
in t and with
a
requested,
as
[0,T]
x
x
E
-»•
Clbr
following properties:
continuous, uniformly bounded by Q
k is
L* and
n(v,t,y)
Kt>y
K2(ni(-,y),t)
:=
[0,T]
3.
Cl'r(R)
k2ok1:
{w
:=
| f k2(v, t) ) (y)\
of y.
introduced Ki and k2
=
such that
way such that k2
continuous in
k
has the
introduce C*'1
us
k2 is constructed in
is continuous
ß'
constant
some
For
and with
if H^'Hoo
< L
Cb'r(R)
:
Lipschitz
k
—
C®'L*
constant
is
|H|oo
and
<
Kh +
(T
^(v,t,y)J(y)\<ß'L
—
t)C
—
t.
Lipschitz continuous, uniformly
independent ofyER.
is
locally Lipschitz
continuous
y G
m
in
E, uniformly
t E
with
[0,T].
to t E
respect
a
v
Cb'r(R) fixed,
E
v
=
KH+CT and
:=
Proof:
ad 1. This claim follows
ad 2. Since
Ki(v,y)
E
C*'L for
\\K2(Ki(vi,y),t)
ad 3. Let
f Ki(v, y)
us
fix
j ||oo
a
-
function
< oo
from the definition of Hi and n2.
directly
any
E
v
Cb'r(R),
K2(Ki(v2,y),t)\\y
Cb'r(R)
E
v
for any y
R.
E
L
as
y£K
We have to show that
t. We know
that,
n(v,t, )
for any y
\\n2(Ki(v,yi),t)
Since this holds for all y
-
E
\\\K2(Ki(v,yi),t)
By construction,
y
i—
:
E
L2(L)\\Ki(vi,y)
<
L2(L)\\\ni(vi,y)
well
-^
as
us
a
compact
Cb'r(R)
is
Lipschitz
continuous
L2(L)\\Ki(v,yi)
<
L2(L)\\\Ki(v,yi)
Ki(v,y)
on
K, uniformly
Ki(v,y2)\\y
-
-
Ki(v,y2)\\\.
<
L2(L)\\\Ki(v,yi)
locally Lipschitz
continuous and
-
Ki(v,y2)\\\.
therefore,
claim is proven.
Appropriate examples
in
get
K2(Ki(v,y2),t)\\\
is
We know that
|| f K\{v, y)
<
-
Ki(v2,
introduce
K2(Ki(v,y2),t)\\y
we
-
set K C R.
K,
R,
Ki(v2,y)\\y
-
R and yi,y2 E
E
from
directly
<
Hence, let
:= max
the claim follows
of functions Ki and k2
can
be found in Section 5.6.
also the third
EXISTENCE OF CLASSICAL SOLUTION
5.4.
Step
rem
The function g
2:
[0,T]
:
x
Ct
l,r
69
fulfills the conditions of Theo-
ch
4.1.6
Theorem 5.4.3
a-2 g
:
[0,T]
x
uniformly
a-3 g
is
The Conditions a-2 and a-3
C,
m
l,r
bounded.
m
4-1-6
continuous, Lipschitz
fixed
For any
uniformly
coefficient equals r
continuous
is
Theorem
fulfilled,
are
continuous
m
i.e.
v
C
E
l,r
[0,T].
t E
uniformly
Holder
Cb(R)
of
y
with respect to
E
v
(t, y)
Cb'r(R), (t,y)
m
gy(t,v) is Holder
compact subsets of [0, T] x E. The
—
1.
—
Proof:
ad a-2
as
well
Continuity of g is ensured since k is continuous.
as y E R, n(v,t,y) belongs to C®'L* for any v
For fixed
Q
:=
Cb'r(R).
E
KH+CT,
For vx,v2
E
L*
=
ß'L
C^'L*
we
may write
\gy(t,vi)-gy(t,v2)\
1
<
fi,t(x)WtL (v2,x)u(dx)
fi,t(x)Wt (vi,x)u(dx)
2*?,
f fitt(x)u(dx)
AM / flt(x)v(dx)
-
a
tL
WtLy(vi)-WtLy(v2))(x)fi,t(x)»(dx)
t,y
i,t
WtLy(vi)-WtLy(v2))(x)u(dx)
By Assumptions 5.1.1,
know that
we
\
as
well
as
f fi(x)v(dx)
—
Xi
f f\(x)v(dx)
are
uniformly bounded on [0,T]. Moreover, we know by Corollary 5.3.4 that WtLy(v) is uni¬
formly bounded for all v E C®'L* by some constant K, and we may write, by using the
b2 < 2max(|a|, |6|)|a
elementary inequality a2
b\,
—
—
\gy(t,vi)-gy(t,v2)\
'
<
f \h,t(x)W(dx)
a
(K
+
l) I \fht(x)\u(dx)
+
Xht I
flt(x)u(dx)
i,t
xllwàM-wàMiu.
The
Lipschitz continuity
the other
uniformly
constants
hand, by
in t
E
of
Wfy
:
C®'L*
-^
is ensured
by
Lemma 5.3.5.
[0,T].
of y, g is
Lipschitz
continuous in v,
uniformly
in t.
ad a-3 Boundedness follows
and 5.3.4. For
let
us
fix
(t,v)
showing
E
continuous in y, if
[0,T]
On
5.4.2.2, ntty : Cb'r(R) — C®'L* is also Lipschitz continuous,
Since both functions are Lipschitz continuous with Lipschitz
Lemma
independent
l°°(su.pp(u))
>l,r
directly from the definition of k as well as Corollaries 5.3.2
continuity with respect to y in a compact set K C E,
Cb'r(R). By the same reasoning as above, gy is locally Holder
the Holder
x
one can
show that there exists
a
constant
WLMv^^))-WUyMv^yM\oo
<
HK such that
HK\yi-y2f
CHAPTER 5.
70
for
a
MARTINGALE MEASURES UNDER ADDITIVE PROCESSES
suitable choice of
ß
(0,1]
E
and any yi,y2
This
K.
E
inequality
can
be shown
as
follows:
\\WtLyi(K(v,t,yi))
W4(^,t,y2))||oo
-
+l|w/i>M>^))
However, for
E
v
know that
n(v,t,y*)
Q
:=
L**
:=
KH
+
sup
\\\K(v,t,yM \
W4(/(w,t,y2))||oo
-
(5.4.5)
term of the RHS of
K,y
E
on
v).
follows
to
< OO,
eR.
Hence, by
<
C\\\\K(v,t,yi)-K(v,t,y2)\\\
-
K(v,t,y2)\\y
5.4.2.3, n(v,t,-) is locally
t, the Lipschitz continuity and hence the
Lemma
proven. The Holder
continuity of the second
The Holder coefficient is
1.
All conditions of Theorem 4.1.6
fulfilled and
are
hence,
u(t,y)
Step
3:
Choose g
:
[0,T]
x
~gy(t,ut)
Step 1,
we
introduced the
the entire function
u
as
well
=
us
E
unique solution
vt,y\
,
ah(Yf'y)+
C61,r(E)
=
a
-
/
V
it to the
Cb([0,T]
E
we
x
may write
boundary
E). Since ß
u(t,y)
in the
7Xs'y
I
gYs"(s,us)ds
such that
(t,y)G[0,T)xE.
auxiliary function g with a truncation function
the slope of u in some neighborhood of point
as
show in the next two theorems that the function
K(ut,t,y)
u
Cb(R) appropriately
gy(t,ut)
apply
we can
problem (5.4.2)-(5.4.3),
gives
and a2 are Lipschitz continuous in [0,T], by Corollary 4.1.8,
Feynman-Kac representation
which
In
5.3.5,
Ci\\K(v,t,Vi)
from Lemma 5.3.6.
directly
Lemma
<
Since, by
Lipschitz continuous, uniformly with respect
Holder continuity for any ß of the first term is
r-
(5.4.5)
CT,
for any y*
C\ (dependent
constant
some
C^'L"
E
Wt>y(n(v,t,yi))
for
K,yMv^yi))\\oo-
Cb'r(R) fixed,
yteK
we
-
k
can
be defined in such
k,
capping
y.
We will
a
way that
=ut-
Theorem 5.4.4
There exist
some
\u(t,y)\
constants
<
KH
Kn,C,t
+
such that
(T-t)C-e.
for
all
(t,y)
E
[0,T]
x
E,
(5.4.6)
EXISTENCE OF CLASSICAL SOLUTION
5.4.
Proof:
Kh
Let
CüH-fflloo,
>
fix t
us
such that
Ty
whereas
Yfy
:=
[0,T],
E
we
inf {s
is defined
define the
u(s,Yfy)
K(us,s,YfyX\
bounded
by
(Yfy)
(T
s)C
-
0. Let
=
constant
some
(T-s)C
+
KH +
>
fUx)
<
C and
constant
time ry:
stopping
+
(T- s)C}
A
T,
Yff)
<
us
&2,udY':
ßudu
y+
_
s
Since
arbitrary positive
as
follows:
as
u(s, Yls'y) >KH
well
as
[t, T] | u(s, Yfy) <KH
E
yt,y
We have
R
E
y
71
for all
for all
E
s
E
s
\t, ry{
\t,Ty{,
show that for all
we
E
s
u(ry,
and
/\K^'Y''V)
get
(T
0
<
-
as
ry)C.
well
\t,Tyl, WFt,y(K(us,s,Yfy),x)
of level C. We know from
independent
KH +
(5.3.2)
as
is
that
(wFt,y(K(us,s,Yfy),x) + l-XiJiyS(x))u(dx)
max
icGsupp(^)
<j
k*(x) \ / \fitS(x)\u(dx)
exp
with
k*(x)
A
=
n(ûa,s,Ys 'y)
i
{x)
_
hAx)\K,(i
1
UMp^l) LAAp^l
+
_
V
J
al,s
al,s
°"2,;
-fhs(xy-^[K(us,S,Yfy))(Yfy)
°"i,.
for
E
s
(^^
l"î
f
\
t
fiAx)
<
AM[
L
V
1 +
$ flÀzXdz)\
f fi,s(zMdz)
I
-2
-2
J
ul,s
Ul,s
{t,ry{. Therefore,
W^t,y«us,s,Yfy),x)
a n(ûs,s,Ya
I
exp<jAoV?,V
'y)'-
s,Ys
1 +
<
and
WFt,y(n(us,
s,
of level C. We then
\
/ h,s(z)WL(z)v(dz)
Ai,
a
l,s
X\sfi>s(x)
exp<^ |/M(a;)| Ai,
1 +
/
r
fiAx)
maxa;esUpp(îy)
\
exp
k*(x)
a
\ f \fiyS(x)\u(dx)
l,s
XhsfitS(x)
Y*'y),x)
can
is
uniformly
conclude from
bounded
(5.4.1)
on
{t, ry{ by
that there exists
some
a
constant
constant
independent
Ci, independent
CHAPTER 5.
72
of
C,
such that
~\rt,y
si1
g
.q
a'
u(t,y)
MARTINGALE MEASURES UNDER ADDITIVE PROCESSES
(s,us)
=
E
<
C\ for all
lt,Ty{.
E
s
We may write
ah(Y^'y)+ / gY'' (s,us)ds
t,v
t,y\
(s,us)ds
E
g"
=
E
gY'' (s,us)ds
<
CiE[ry-t]+KH
vl,y
-.Y,
gs
(s,us)ds
+Eu(Ty,Yff)
.
+
t,v
/
+ ah(YT'y) +
CE[T-ry]
due to the strong Markov property.
Let
us
consider the lower bound.
from below.
[0, T]
x
Hence, there
E. If
exists
fix C >
we now
C\
the definition of
By
a
constant
V
C2,
C2 such
us now
show that KH
can
some
constant
Let
e.
us
start with
all
(t,y)
E
Kh
a
way that
(T-t)C-t
+
:=
(T-t)C.
+
be fixed in such
\u(t,y)\<KH
for
is bounded
may write
we
\u(t,y)\<KH
Let
gy(t, ut), we know that g
that gy(t,ut) > —C2 for
a\\H\\00
+ 2t. We know
by
the definition of
g that
~9y(t,ut)
for
some
positive, increasing function K(x) with K(0)
u(t,y)
Let
<k(kh + C(T -t)- u(t,y))
us now
7.Y''y
directly
E[ah(Yf'y)+
~gY°'
<
a\\H\
K(KH
<
aWHW^
+
KiKH-aWHW^iT-t)
<
a\\H\\QO
+
K(2e)(T-t).
E
see
we
may write
,
(s,us)ds\
+
C(T
-
s)
-
u(s, Yfy))ds
K(2e)-C
that
therefore, g(t,ut)
a\\H||oo
Hence,
define
u(t,y) ^all^Hoo+ C(T-t)
and
C.
=
T' :=T
We
>
+ 2e. Let
=
g(t,ut)
us now
u(t,y)
fix Kh
on
'=
+
for all t
e
G
[T',T]
[T',T]. However, this holds for any Kh with Kh
a^H^oo + C(T T') + 3e. We know that
—
C(T'-t)
<
KH
<
a\\H\\QO
+
+
C(T-T')
+ 3e +
C(T'-t)
>
5.4.
EXISTENCE OF CLASSICAL SOLUTION
for all t
E
[0, T'].
On the other
hand,
we
73
also have
on
[0, T']
T
u(t,y)
gY°'y(s,us)ds
<
Eu(T',Y^y)+j
<
all^Hoo
+
C(T -T')
+
t
+
k[ku
<
a\\H\\00
+
C(T-T')
+
t
+
K(2t)(T' -t).
We conclude that for T"
u(t,y)
<
T'
:=
«Halloo
-
C(T
-
-
T')
-
e\ (T'
-
t)
K/2%-c^
—
a||Jff||00
C(T-t)
+
for all t
+ 2e
G
[T",T].
By iteration, we generalize this reasoning to T^n\ Let us now define k the smallest
*- e'~
T. By the above reasoning, we conclude that for Kh '=
integer such that k >
a\\H||oo
+
(k
+
l)e,
we
get
u(t, y) <KH
Since
we
+
C(T -t)-t
have finished the
Theorem 5.4.5
Ki(ut,y).
inequality (5.4.6)
that
proof
There exists
a
We know from Theorem 5.4.4 that
Let
us
define L
>
ry
-§-u(Ty,Yf.'y)
:=
on
we
<
[0,T].
for
all
inf{s
G
(t,y)
E
[0,T)
E,
x
(5.4.7)
L.
may choose
(Kh, C, e)
aLH with LH being the Lipschitz
and
< L
holds
constant L > 0 such that
tion 5.1.2. We introduce the time ry
We have
[0, T].
-aWHWoo-CiT-t),
>
\^u(t,y)\
Proof:
E
have
u(t,y)
we
for all t
such that
n,(ut, t, y)
constant from
=
Assump¬
as
[t,T] | —u(s,Yfy)
dy
-ß-u(s,Yfy)
> L
for all
s
<
E
L}
A T.
\t,ry\,
which result in
dy
_
d_ ki(us,Yfy)(Yfy
dy
f-Yty+h(x)
ß
dy
Ui'y
(which gives
=
which is
negative
on
f2(x))
-
Ki(us,Yfy)(Yfy))
Ki(us,Yfy))'(z)dz
due to
(5.4.4))
{ni(us,Yfy))(Yfy
\t, ry{
+
+
f2(x))
-{ki(us,Yfy))(Yfy),
since
Ki(us,Yfy)) (Yfy)
=
L
(5.4.8)
CHAPTER 5.
74
MARTINGALE MEASURES UNDER ADDITIVE PROCESSES
and
Ki(us,Yfy))
x
E
supp(i/). By (5.4.8),
Since ut
E
C
for all
1,1
in y, and
uous
we
hence,
(Yfy
it is differentiable in y
d
,
dy
dy
^
ah(Yf'y)+ /
d_ E
vl,y
g
dy
=
E
v
E
Cb (E),
ß-9y(t,v)
=
let
us
E
t
ß
t,v
s
g
dy
-§-gy(t,y),
0
on
{t,ry{.
locally Lipschitz
is
contin¬
.
(s,us)ds
d
y
L + E
=
(s,us)ds
dy^^X
discuss
xy!«
gr°
d
ry
<
s
(Yfy)
We may write
a.e.
^
E
< L
gy(t,ut)
know from Theorem 5.4.3 that
<9_
u(t,y)
For any
f2(x))
(Ki(us,Yfy))
also get
we
+
-w'
(s,us)ds
~Y^,
—g
oy
s
(s,us)ds
which is defined
a.e.
We
get
—gy(t,Ki(v,y))
f fi(x)WyL(Ki(v,y),x)u(dx) [
/
L
{Ki{v,y),x)v(dx)
/ fnx)-Q~Wy
°l
+
d
d_
^WyL(Ki(v,y),x)u(dx)
dy"y
with
WyL(x)
Since
in y
Ki(v,y)
a.e.
+ 1
-
»1,1
XJi(x)
(E), Wy (ki(v,y),x) is Lipschitz
By differentiating equation (5.2.10) applied
E Cb
d
=
continuous and hence differentiable
with
Ki(v,y),
we
conclude
{d-yAyliV^
-fi(x)
we
> 0.
d
dyWy^
Hence,
WyL(x)
:=
o2(
,
,\»,
—[Ki(v,y))
Lai
s
(y)
,
+
f
f\(z)lWyL(z)u(dz)
°l
WyL(x).
follow
d_
WyL(x)u(dx)
dy"y
d
dy
o-2
A^>y\x))WyL(x)»(dx)
-
y
^(Ki(v,y))
o\
f fi(x)WyL(x)u(dx)
CÎ
d
fi(x)^WyL(x)u(dx),
(y)
fi(x)WyL(x)u(dx)
OPTIMAL MARTINGALE MEASURE VERIFICATION
5.5.
such that
may write
we
JV(M)
=
j {ly^y\x))w^y(x),(dx)
'rci(t>,y))"(y) j fi>t(x)Wfty(x)u(dx)
uu
and
75
therefore,
on
the set
[t,ry[,
(5.4.9)
get
we
—
dy
~gYl'y(s,us)
<0,
respectively
—u(t,y)
The lower bound
can
be shown in
exactly
L.
<
the
same
way such that
\-^u(t,y)\<L.
In this
section,
we
may write
we
M
have shown that there exists
unique function
a
u
Cb([0,T]
E
x
E),
boundary problem (5.2.12)-(5.2.13). By Corollary 4.1.8, we may write
the solution in the Feynman-Kac representation (5.2.14). In the next section, we will
discuss the remaining missing element of the proof of Theorem 5.2.1. We will show that
u truly defines the optimal martingale measure.
which solves the
Optimal Martingale
5.5
We finalize this section with the
martingale
is defined
Q
measure
QB.
measure
Let
(pB
=
that the
-^—(p
resulting strategy
and °B
=
~~
^(0,0),
u
defines the
such that the
rT
'
exp|ci?+
dP
/
(J)fdSht
+
aB\.
perform
the Verification Procedure outlined in Section 2.4. We
in
also the restrictions of
reasoning
a
that
Let
remind that
us
1.
Q
is
Let
we
an
us
defined
general set-up, capturing
/i to be random; however, they
(p, aL and WL are uniformly bounded.
more
means
allow <Ti and
equivalent probability
must be
provide the
Chapter 6. This
even
uniformly
We consider the local
martingale
as
=
=
-
j XidM
f(aL
-
bounded.
measure:
check the conditions of Lemma 2.4.1.
N
optimal
probability
by
dQ
We
proof,
Measure Verification
+ L
Xiai)dYc
+
(wL(x) Ai/i(x))
-
*
(ßY
-
vy).
N
CHAPTER 5.
76
Since
WL,
MARTINGALE MEASURES UNDER ADDITIVE PROCESSES
Xi and /i
bounded,
are
N is
bounded and due to
locally
WL(x)-Xifi(x)>-l,
have AN
we
>
U
—
1.
Moreover,
\
l(oL
=
+
we
set
Xioifdt
-
[1 XJi(x) + WL(x)j log (WL(x)
+Xifi(x)-WL(x)X*ßY.
\
-
Xifi(x)
-
+ 1
Ai, (p, oi, aL, fi as well as WL are all uniformly bounded,
locally integrable variation and its compensator is bounded as well. Hence,
condition (2.4.2) is naturally fulfilled and therefore, Q is an equivalent measure. At
the same time, since L is locally bounded, we get due to Corollary 3.2.2, that Q
Since
v(R)
<
and
oo
U has
belongs
2.
Q
is
By
a
the
L and
to M.e.
martingale
same
3.
I(Q, P)
The
argument
[M, L]
martingale
measure:
as
bounded.
locally
are
in Section
3.4, WL
Hence,
is bounded and
get due
we
to
therefore,
Corollary
the processes
3.2.2 that
Q
is
a
measure.
< oo:
density
Z
-Jp
=
may be written
as
rT
Z
where cB is the
I(Q,P)
=
exp
=
normalizing
Eq cB
< c
B
/o
Si,t
constant. We
4>t
+
Si>t-
>o
-dSi
t
4>t
-dSi
t
+ a.B
get
+ oiB
rT
=
E-,
Q
cB +
/
Mvi,tdt
<JhtdYtc)
+
(f)fi(x)
+
*
(ßY
-
iv)
+ aB
We therefore have to show that
4>t
EQ
c
+
then, I(Q,P)
Assumption 5.1.2, ensuring
since
=
,Q
v'$
we
s i-,
-dSi
are
local
=
(5.5.1)
0,
Eq[(xB]
which is finite
that the
contingent claim
=
(WL(x)-Xifi(x)
get by Girsanov's Theorem that fi(x)
aL)aidt
;
Q-martingales.
In
*
(ßy
+
—
fact, they
by
the previous step
B is bounded.
as
well
as
Introducing
l)*vY,
Vy)
are
as
well
true
as
/ oidYc + f(XiOi
martingales
—
since their
EXAMPLES OF Kl AND k2
5.6.
quadratic
variations
be written
are
77
Q-integrable. (5.5.1)
follows since the
dS
^~
(TittdYtc
=
(\itt<Ti,t
+
-
/
(rf)(Tittdt +
d(fi(x)*
v
<~>i,t-
4.
f
(p
is
such that
uniformly bounded,
-çr—dSi
is
for all
Q-martingale
true
a
f -^—dSi
Q
E
can
positive
constant
a
-
.Q"
Vy)
/*
martingale.
true
preparation, let
us
observe that for
have
a we
E
is
(ßY
M.e with finite relative entropy:
We will check the condition of Lemma 2.4.2. As
any
of Si
as
——
and
dynamics
<^ afl (x)
exp
*
ßY
<
oo,
(5.5.2)
<
oo.
(5.5.3)
rT
E
The first
/
a1
expl (afl(x)
*
n
ßY
(1992),
He et al.
e.g.
Lemma 2.4.2 if
we can
E
We denote k
inequality
exp<j/3 /
v
<Pt
Cb'r(R)
for
L for all
x
E
< oo,
follows since o\ is
r
G
>
0 and
R with
us
take
d[S]u
\x
—
Q-martingale
< oo.
C2:2
Let
true
ß
=
p-.
the
By
Cauchy-Schwarz
get
< E
exP
i
/
o\,tdt +
yfl(x)
*
ßY; t
< 00.
of K\ and ki
(1, 2].
a
ß
a
0
>
-F^d[S]i}t
\ß I
we
some
ySitt-
A reasonable
kAvu)(x)-{
for any fixed #1
exP
It will be
Q-martingale.
^2
(77^—)
Examples
E
local
a
show that for
(5.5.2), (5.5.3)
and
o
Let
is
supt[0T] ||0||oo-
=
T
5.6
l)»(dx)dt
Inequality (5.5.3)
Lemma 14.39.1.
f -^—dSi
It is obvious that
E
-
bounded.
uniformly
by
eaf2^x)
JS.
'0
see
tdt
follows from
inequality
E
exp
< a
V{X)
suitable function
y\
<
11/21100-
example
if
of «i is
|^-y|
§i(v,y)
E
C(
>
l,r
||/2|U
+
^1
such that
®i(v,y))
(x)
<
CHAPTER 5.
78
We
that $1
assume
MARTINGALE MEASURES UNDER ADDITIVE PROCESSES
on
{x
E
\ \x
R
y\
—
H/2II00
<
I
$i(v,y)(x)
v(y)
=
+
I
Si}
+
I
II
is of the form
II
£
nrv
\x-y\- H/2II00
a
v'(z)dz
Si
\x-y\- II/2II00
a
—
v'(z)dz,
Si
whereas
•
E
a :
—
[0,1]
C2(R)
G
is
increasing function, such that
an
a(x)
a'(0)
0,r-l
v'ec
conditions,
Hi
Ci
:
0,
=
a"(l)
0,
=
l,r
x
v'(z)
sign(v'(z))L
:=
E
—
iï\v'(z)\ <L,
if ^'(z)! > L.
C6'r(E), defined
with above
$1, fulfills
the necessary
i.e.
&Ki(v>y)(.x)
L
=
1,
=
with
v'(z)
Lemma 5.6.1
a'(I)
=
a"(0)
0, a(l)
0 Vx<
=
<
for
L
sign(v'(y))L if \v'(y)\
all
{x
E
x
L, Ki(v,y)
=
E
=
| |x
R
v
y|
-
if H^'Hoo
H/2II«,}, Mv,y))'(y)
<
<
L,
and condition
(5.4-4)
=
%s
fulfilled,
2.
v 1—
3.
for
Ki(v,y)
v
E
Proof: Let
,
+
Cb'r(R),
us
#1},
is
we
first
Lipschitz continuous, uniformly
y
Ki(v,y)
ensure
$i{v,y)){x)
l$(v,y)j
and
($(t>,y)) (y
±
us now
ad 1. The
Cb'r(R).
E
sign(x-y) ,{\x-y\
=
(II/2IU
«
;
V
Si
k-y|
continuous.
We know that
are
+
uniformly
Si))
-
-
II/2II00
=
v'(y
r
—
bounded
±
1.
(||/2||oo
Hence,
||/2|
{x
on
v'(z)
£1
v'(x)
Si
Holder continuous with coefficient
Let
locally Lipschitz
Ki(v,y)
that
-a
&(v,y)
is
y,
E
\ \x
R
—
y\
<
have
un
*/
1—
m
on
+
{x
Si))
we
—
v'(x)
E
as
R
+
\ \y
well
as
conclude that
show all the above claims:
properties follow directly from the definition of
«i.
—
v'(z) )dz
v'(x).
—
x\
H/2II00
<
($(v,y))'
is
Ki(v,y)
Cbr'
E
+
^1},
obviously
5.6.
EXAMPLES OF Kl AND k2
79
ad 2. We have to show that
\\\Ki(vi,y)
for all vi,v2
G
;l,r-
Cb'(R) and for
For obvious reasons,
\Ki(vi,y)(x)
<
-
—
a
-
v2\\\
{x \ \x
—
y\
H/2II00
<
Si}. Hence,
+
Ki(v2,y)(x)\
\vi(y) -v2(y)\
I
Lv\\\vi
<
suitable constant Lv:
a
consider
only
we
Ki(v2,y)\\\
-
\x-y\- II/2II00
+a
v'i(z)
Si
\x-y\- H/2II00
-v'2(z))dz
v'i(z)
Si
-v'2(z))dz
Since
\v[(z)-v'2(z)\<\v[(z)-v'2(z)\,
we
get
\Ki(vi,y)(x)
Ki(v2,y)(x)\
-
<
\\vi
-
w2||oo
+
2( ||/2||oo
+
) IK
<*i
-
v'2
On the other hand,
Ki(vi,y)j
-
12
2||c/||(l
v[(z)
\v[(x)
Si
3 +
(x)
00
\x-y\- II/2II00
-a
(
\Ki(v2,y)j
-
,(\x~y\
1
^
<
(x)
12
-
v'2(z)dz
—
v'2(x)\
+
\v[(z)
v[(z)
-
v'2(x)\
-v'2(z))dz
\v[(x)
+
-
v'2(x)\
oo
^2lloo)
\V\
Si
whereas
\a'\\
:=
sup
[0,1]
|c/(t)|.
te
Therefore,
the second claim is proven.
ad 3. Let
v
\v'(y)
—
>l,r-/
E Cb'r(R)
v'(z)\
<
hft\y
claim is proven if
such that for any compact set
—
z\r~l
for all y,z
E
K.
K, there
Let K be
a
y£K
x
d
sup sup
y£K
x
dy
d_ Ki(v,y)(x)\
dy
Ki(v,y)j
(x)\
h^
< 00
such that
fixed compact set in E.
we can ensure
sup sup
exists
<
00,
<
00.
The
CHAPTER 5.
80
For obvious reason,
MARTINGALE MEASURES UNDER ADDITIVE PROCESSES
we
^i(v,y)(x)
consider
only
[y
E
x
H/2II00
—
us now
Si].
+
We may write
||/2|
-
(v'(z)
Si
-
v'(z))dz,
2l^(Si
<
2\v\y)\
<
2fl + (l + ^
+
\\f2U\\v>\
+
U )\\v
u
\\a
oo
/ II
||oo-
consider
d
dy
(*i(v,y))'(x)
1
sja
=
„(\x~y\
CO
(v'(z)
>
(
v'(z))dz
-
jy
,(\x-y\
-a
Si
we
12
Si
sign(x-y)
such that
II/2II00
y +
get
\^i(v,y)(x)\
Let
y\
-
-OL
Si
we
Si,
[l-a[lx yl6m°°))(v,(y)-v'(y))
=
sign(x-y) ,(\x
such that
—
\\f2\]00
-
w
Si
,
) ( v
_,
(y)
v
-
(y)
,
-
v
(x)
+
_,
v
(x)
>
,
get
d
dy(<£i(v,y))>(x)\
<
fj\a"\\(Si
+
H/aUIKIU
+
^||a'|||t/(y)
-
v'(x)\
whereas
W(y)-v'(x)\<hk(Si
\\f2\\00)
+
r-l
and
K
is
a
compact
Hence,
y
1—
:=
is
example
'
K2(v,t)(x)
=
some
KH +
-
z\
<
12
S
»
uniformly
bounded
on
K.
continuous.
of k2 is
v(x)
<p2{v,t){x)
if
e
<i
\v(x)\ <KH + (T -t)C-t
+ (T-t)C-e<\v(x)\
<KH + (T- t)C
iîKH
<
fixed constant
(T- t)C.
\y
such that
K
locally Lipschitz
sign(v(x))^KH
for
E
We conclude that also the second term is
set.
Ki(v,y)
A reasonable
\z ER\3y
+
(T-
Kh and
a
t)c)
suitable
if
\v(x)\
ip2(v,t)
>
E
KH
7l,r,
+
(T
-
t)C
Cb'r(R) with
\ip2(v,t)(x)\
<
EXAMPLES OF Kl AND k2
5.6.
We introduce
be defined
k(t,x)
KH +
:=
81
(T-t)C + x.
Let tp2
[0,1]
:
[0,1]
—
k(t,0)
C2(R) being increasing,
G
/3(0)
/3"(0)
Lemma 5.6.2 k2
:
C6'r(E)
x
[0,T]
—
C*'1
:
with
Proof:
a
Let
[0,T]
x
—
Lipschitz
us
first
C*'1
is
0.
all necessary
conditions,
constant
ensure
—
t)C) if \v(y)\
t.
In
continuous
function
mv
E
K2(v,t)
>l,r,
E Cb
(E).
We have
Holder continuous with coefficient
Q
=
on
{x
on
{x
Hence, K2(v,t)
G
(x)
<
\v'(x)\
on
E
ER
R
r
1. We have
—
| \v(x)\
I \v(x)\
=
=
k(t,-t)},
k(t,0)}.
{x ER\ k(t,-e) <\v(x)\<k(t,0)}.
C61,r(E).
prove the other claims:
ad 1. These
ad 2. Let
us
properties follow directly from the definitions.
show that
llv^i,*)
for ^1,^2
G
C*'1 and
a
-
^p2(v2,t)\\y
suitable constant
\tp2(vi,t)(x)
-
<
L2(l)\\vi
-
v2\
L2(l). Obviously,
tp2(v2,t)(x)\
addition,
C*'1, uniformly
L2(l), independent of y.
that
=
ß'.
suitable
some
-
i.e.
as
<P2{v,t)J
us
k(t,0)}
1,
—
Lipschitz
a
'<p2{v,t)\{x)
Let
ß"(l)
=
=
ß'\v'(y)\ for
<
( ip2(v,t) ) being
well
<
1,
=
C6'r(E) fulfills
ip2(v,t)j(x)=v'(x)
as
=
=
K2(v,t)\ (y)\
with
=
-
k2
\v(x)\
k(t,0)
IMM)Hoo < KH + (T
t)C, K(v,t)(y)
sign(v(y))(KH + (T
and
v
+
Kh
if ||w||oo < KH + (T
(T
t)C
K2(v,t)
t)C
—
2.
<
such that
ß'(l)
0,
=
\v(x)\
-
0, /3(1)
=
/3'(0)
1-
\ k(t,-t)
R
ß[Ht,o)-\v(x)\m_e)
sign(v(x))
=
l-ß
ß
E
as
tp2(v,t)(x)
with
{x
on
<
\vi(x)
-
v2(x)\.
m
t
82
CHAPTER 5.
MARTINGALE MEASURES UNDER ADDITIVE PROCESSES
On the other hand
V2(vi,t)) (y)
<
-
(ip2{v2,t)) (y)
v[(y)ß'(k^0)-lVl{y)l)-v'2(y)ß^k^0)
~
M?/)l
t
<l
ßlfk(t,Q)-\vi(y)W ^/fk(t,0)-\v2(y)\
t
\vM-v2{y)\
<L
+
t
\\nK{y)-^{y)\
with
r\\
Hence,
k2 is
Lipschitz
continuous
on
:=
CI'1.
sup
te[-i,i]
\ß"(t)\.
Wi(y)
~
v'2(y)\
Chapter
6
MEMM for Stochastic
Volatility
Models
In the literature
part of the
stochastic
(1991),
mathematical
on
examples
volatility
Heston
for
finance, stochastic volatility models provide a major
incomplete financial market models. While most of the original
models
(1993)),
are
set up in
some newer
a
Brownian motion framework
models allow for
jumps
in the asset and the
A prominent member of this second class is the stochastic
process.
(Stein
volatility
and Stein
volatility
model of
Shephard (1999, 2001), hereafter termed the BN-S models. The
corresponding volatility process is constructed via a mean reverting, stationary process of
the Ornstein Uhlenbeck type driven by a subordinator, a Levy process with no Gaussian
component and positive increments. In detail, such a model can look the following way:
Barndorff-Nielsen and
JO
-^
=
St-
da2
where the parameters ß,
=
ß, X,
motion while the process
with the
with
are
an
so
Z\
{ß
+
ßa2_}dt
—Xa2_dt
p
=
are
+
+
+
(epx-l)dZxt
dZ\t,
real constants with A
(Z\t)te[o,T]
is
called
increase
at-dBt
a
>
0 and p < 0. B is
subordinator.
a
Brownian
This model allows to deal
leverage type problem, i.e. for equities a fall
in future volatility. The Brownian Motion B
in the
price
is associated
and the subordinator Z\
independent.
hedging for stochastic volatility models has already been analyzed by several
Biagini, Guasoni and Pratelli (2000) discuss mean-variance hedging for stochas¬
tic volatility models, where the volatility process may have jump elements. The MEMM
for continuous stochastic volatility models has been analyzed by Rheinländer (2005) as
well as Hobson (2004). Grandits and Rheinländer (2002) and Benth and Meyer-Brandis
(2004) determine the MEMM in stochastic volatility models where the price process is con¬
tinuous and driven by a Brownian motion B, whereas the volatility process may contain
jump terms and is orthogonal to B. Still assuming a continuous price process, Becherer
(2004) considers a model with interacting Itô- and point processes. The main contribu¬
tion of this chapter is the determination of the MEMM in a general class of stochastic
volatility models with discontinuity encompassing the simpler case where the price process
is continuous with an orthogonal pure jump volatility process. Our approach presents a
unifying framework which moreover allows to handle models like the BN-S model. We
Pricing
and
authors.
83
CHAPTER 6.
84
provide
MEMM FOR STOCHASTIC VOLATILITY MODELS
the MEMM in the form of the solution of
tion, similar
to the
optimal martingale
an
partial
interactive
differential equa¬
Chapter 5.
leverage effect, Nicolato and Venardos (2003) an¬
alyze the class of all equivalent martingale measures, then concentrate on the subclass
of structure preserving martingale measures (i.e., the price process is also of BN-S-type
under those martingale measures). Structure-preservation is very suitable for analytical
purposes, however, it is often not warranted in practical applications. While in case of
exponential Levy processes, the asset process under the MEMM is again an exponential
Levy process (see in particular Esche and Schweizer (2005)), one major implication of this
chapter is that the volatility process in the BN-S model in general has no longer indepen¬
dent increments when seen under the MEMM. Therefore, only considering the class of
measure
in
With respect to the BN-S model with
preserving martingale measures seems to be
the concept of exponential utility maximization.
structure
in
This
is structured
chapter
assumptions for
and in
introduce all necessary
we
stress the similarities and the differences with the
Assumptions
again restrict the Levy process Y to the case where v(R) < oo, i.e. Yd represents
compound, compensated Poisson process. Let us now consider a class of stochastic
Let
a
approach, especially
a
Model and
6.1
section,
narrow
approach introduced in
Section 6.3, we present the main elements of the proof, arguing again closely
In Section 6.4, we provide some comments with respect to the independent
Sections 6.5 and 6.6, the solution to the BN-S model is provided.
especially
Chapter 5. In
to Chapter 5.
case
In the next
too
generic stochastic volatility asset model. In Section 6.2, we motivate
partial differential equation and present the main theorem of this section.
the interactive
We
follows.
as
a
us
models
volatility
with V
such
a
^
=
r!(t,Vt-)dt
dVt
=
rf (t,Vt-)dt
values in
taking
and
+
3.
+
d(wM(- ,V_,x)*(ßY-vY))t
d(wv( ,V-,x)*[j,y)
interval E C E.
(6.1.1)
(6.1.2)
,
We often abbreviate the processes in
Wt(x)
instead of
are as
follows:
we
rj(t, Vt-), a(t, Vt-)
and
W(t, Vt-,x)
6.1.1
coefficient r]v is differentiable m y with bounded Lipschitz continuous derivative
locally Lipschitz continuous mt E [0,T]. Wv is differentiable m y with bounded
derivative and continuous
2.
<jM(t,Vt-)dYtc
+
some
price models of the following type:
write rjt, ot and
Assumption
The
asset
Our basic assumptions
way that
respectively.
1.
consisting of
coefficients
differentiable m
The
is
positive and
WtM(y,
)
supp(u)
WM and Wv
are
are
also
locally Lipschitz
with bounded derivatives
tive, aM
:
t.
aM and WM
r\,
y
m
m
—
on
[0,T]
x
E.
continuous
Further,
m
r\
is
t
and
posi¬
uniformly
from zero on [0,T] x E, and
(—l,oo) is uniformly bounded, uniformly on [0,T] x E.
Gioc(ß).
bounded away
6.2.
MAIN THEOREM
85
4V
T.=
'
uniformly
is
bounded
(aM)2
[0,T]
on
x
f(WM(x)fv(dx)
+
E.
Remark 6.1.2
1.
Assumptions 6.1.1.1-6.1.1.3 ensure due to Protter (1990), Theorem V.38 and the
remark following it
that there exists a unique solution (S,V) to equations (6.1.1)
and (6.1.2) which does not explode m [0,T].
-
-
2.
One
can
easily
Vt
of.
It
=
is
see
that the above model
a2
the set
E,
of
is
values
e~xta2
=
+
=
6.2
The
get
an
equivalent
[<7o,oo),
stochastic
eÄsdZXs
volatility
>
al
(S,a2),
model
such that all above assumptions
with a2
having
values
m
fulfilled.
are
Main Theorem
density
of the MEMM
Q°
can
be written
dQ°
The functions
S_(TM and
a
and W
(x)
of
as
exp^cu+ /
dP
to
\fVt.
eXta2
CTn
E
/ eXsdZXs.
Xt
e
ofVt, would therefore be (0,oo), which results m ajr
from zero. However, by the variable transformation
o+
=
the BN-S model with
not bounded away
being
we
generalization of
a
well-known that
Chapter
3
<j?tdSt\.
with
correspond,
S_WM(x), respectively. Introducing Xt
fundamental equation
(3.3.3)
can
therefore be written
:=
a
slight
abuse of notation,
St-Xt and (pt
:=
St_0°,
the
as
rT
M\2
(o-f-Xto-n2+4>tXt«)
WtL(x)
0
-
(cPt
+
dt
Xt)Wf(x)
+
<PtXt(Wf(x))2)u(dx)dt
JR
rT
aï-^t
+
log(l
-
+
XtW dYt
XWM(x)
+
WL(x))
-
(PWM(x)
*
ßY
(6.2.1)
CHAPTER 6.
86
In
addition,
we
MEMM FOR STOCHASTIC VOLATILITY MODELS
(3.3.2)
know from condition
of
that
and
WtL(x)
must be defined in such
way that
a
tM
o?oLt+
Wr(x)Wf(x)v(dx)
(6.2.2)
Q.
=
As in Section 5.2, the jump times of Y may, because of the finite Levy
counted in increasing order 0 =: ro < Ti <
such that we can write
•
•
measure
v, be
•
,
log
1
XWM
-
£ [log (l
(x)
+
WL(x)
ArîWrM(AyrJ
-
(PWM (x)
-
ßY
T
W£(AYrS)
+
-
kWTM(AYTJ
lr,<T-
t=l
Differently to Chapter 5, we cannot argue that the contingent claim B is exposed to
the jumps of the volatility process V. However, the situation is insofar the same as the
volatility process is not traded; hence, this process has similar properties as the return
process X2 of the non-traded asset in Chapter 5. Therefore, we apply the same idea and
introduce the notation
&ly(x)
:=
u(t, y
We make the Ansatz that there exists
log
(l
-
XtWtM(x)
+
a
Wv(t, y, x))
function
Wf(x))
+
-
u(t, y).
such that
u
$tWtM(x)
-
=
AlVt_(x),
(6.2.3)
with
u(T,
as
well
Taking
as m
is
a
sufficiently
.
)
=
(6.2.4)
0.
smooth function.
into account that there
u{Tt+i,VTt+1-)
=
jumps of
are no
-
Y in
u(t%, VTi.
(rt,rt+i),
we
get
WY(xTJ)
+
u(rl+i, K.+1-) -u(rt, VTJ
du(t, Vt)
(n,n+i)
~-u{t,VtJ)
+
'(n,n+i)
We may therefore rewrite
c°
+
equation (6.2.1)
riY--yU{t,VtJ)
dt.
as
u(0,Vo)
1
2(aB
vt
~
-
\to?)2
+
<PtXt(o^)2
+
/ [WY(x)-((Pt
(<k
+
At)<7.
dYtc.
+
d
+
jtu(t,
Xt)WYI(x)
Vt.)
d
+
VÏWu(t, Vt.)
M,
+ (PtXt(Wr(x)Y)iy(dx)
dt
(6.2.5)
MAIN THEOREM
6.2.
To
87
(6.2.5)
that the RHS of
ensure
+
well
we now
require that
+
+
+
+
(6.2.2)
as
constant,
\(o-L XaM)2 fX(aM)2 !«(., V.) ^«(«, V.)
j (wL(x)-($+X)WM(x) fX(WM(x))2y(dx)
-
as
is
=
(6.2.6)
0
in combination with
aL
By equation (6.2.7)
=
($+X)aM.
in combination with condition
T
<P
(6.2.2),
we
get
fWM(x)WL(x)u(dx)
v
\^ i/\u,^
>
J
V
=
(6.2.7)
-X
—^-2
LeoQ\
(6.2.8)
,
(aMy
which, replaced
in
equation (6.2.3), leads
WL(x)
-l +
We
see
and y
Vt..
9yit,ut)
Let
us
:=
us
Theorem 6.2.1
compact
1.
Let
and aL
+
itXt{^)2
are
dependent
on
ut
:=
u(t,
)
:
E
—
E
f (Wf(x)
(6.2.10)
($t Xt)wtM(x) ^tXt{WtM(x))2)y(dx).
+
-
+
result to Theorem 5.2.1 for the
corresponding
case
of Stochastic
Models:
Volatility
a
(6.2.9)
(p
\{af -Vf)'
state the
now
J\WM(x)
introduce
+
Let
[A+^
XWM(x).
that WL and therefore also
:=
v
'
v
'
y
fWM(z)WL(z)u(dz)
exp|A»-
=
to
(Main
set such that
us
Theorem
aM
consider the
-u(t,y)
+
is
II)
Let
uniformly
Assumption
bounded
on
6.1.1 be
[0,T]
x
in
place.
Let E C E be
E.
boundary problem
r]v(t,y)—u(t,y)+gy(t,ut)
=
0,
(t,y)
u(T,y)
=
0,
yEE,
gy(t,ut) being of the form (6.2.10). Then,
Cb ([0,T] x E), which may be written as
with
u(t,y)=
there exists
gV°'V (s,us)ds
with
Vfy
=
y +
E
J\v\u,V^)du.
[0,T)
x
E,
(6.2.11)
(6.2.12)
a
unique solution
u
E
(6.2.13)
CHAPTER 6.
88
2.
Having
introduced
u
MEMM FOR STOCHASTIC VOLATILITY MODELS
above,
we
fix
u
Let
u.
=
(p and WL as defined m equation (6.2.8)
martingale measure Q, defined by
with
^-exp{c°+
dP
with c°
—u(0,Vo),
=
is
the minimal
us
introduce the
and
strategy
0°
:=
-^—(pt
(6.2.9), respectively. Then,
the
C$dSt
entropy martingale
measure.
Remark 6.2.2
1.
Since the involved interactive
rem
4-2.3 with the weakened
diffusion term m the volatility process. We conjecture
that such an additional term could be included, however, at the expense of additional
assumptions on the functions r]v, rj, aM and WM. Since we do not have the diffusion
element m the original BN-S model, we did not include the diffusion element m the
volatility process.
reason
2. E
for
partial differential equation is of first order, Theo¬
Assumptions b-l-b-3 can be applied. This is the mam
being
models.
solution
3.
As
m
not
considering
respectively aM being bounded is not consistent with BN-S
However, the MEMM m case of BN-S models can be still determined by the
u of the boundary problem (6.2.11)-(6.2.12) as will be seen m Section 6.5.
set
compact
a
Chapter 5,
equation
a
we
(6.2.7), aL
may describe the entire
is
well
density
process Z
defined by (p. Having (aL,WL),
=
the
(Zt)te[o,T\- Taking
density
process
is
entirely defined.
4- The result might be easily extended
ity
T,
at time
B
=
h(Vr) by changing
the
u(T,y)
However, h
must
fulfill
condition c-3
The next section is dedicated to the
central
contingent claims
to price
proof
boundary
=
the value
of
volatil¬
condition to
h(y).
Theorem
m
on
4-2-4-
of Theorem 6.2.1.
checking the existence of a classical solution to the boundary prob¬
lem (6.2.11)-(6.2.12). Differently to the parabolic situation, by Theorem 4.2.3, there is
in general only a weak solution u E Cb([0,T] x E) to the PDE (6.2.11) with boundary
condition (6.2.12). This part of the proof is closely related to the proof of Theorem 5.2.1
and hence, we will only present the main elements. To ensure that m is a classical solution
to the boundary problem, we will need additional arguments as we know from Theo¬
rem 4.2.4.
Showing that u truly defines the MEMM is already shown in Section 5.5 and
problem
lies in
As in Theorem 5.2.1, the
will therefore be omitted.
6.3
Let
us
Proof of the Main Theorem
first rewrite
(6.2.10) using (6.2.7)
and
(6.2.8)
as
PROOF OF THE MAIN THEOREM
6.3.
89
fWM(x)WL(x)u(dx)
g(-,v)
=
-
M
~
.
)
A
-
((7
M
)
a1
fWM(x)u(dx)-Xf(WM(x))2u(dx)
,
'
W
(o-My
j WL(x)u(dx)
+
As
In
already
analogy
noted in Remark
5.4.1,
of
WL,
^(E1)
—
Lipschitz
is not
However, differently
not ensured that
step, in which
u
we
to the situation in
u
Definition of the
appear.
show that there exists
Chapter 5,
it is
is differentiable in the space variable.
show that
we
|^
terms of the form
no
need not be controlled. In three steps,
1:
Cb(E)
x
(6.3.1)
continuous.
proof of Theorem 5.2.1, an auxiliary function g will be chosen to cir¬
problem. However, the function g is much simpler due to the fact that in
defining equation (6.2.9)
Step
f(WM(x))2u(dx),
A2
[0,T]
:
L
(x)W (x)u(dx)
to the
cumvent this
the
g
-
M
only a
Hence,
So,
the
slope
of
v
solution to the PDE.
a
weak solution since it is
we
have to add
a
fourth
is also differentiable.
function g
auxiliary
[0,T]
:
x
Cb(E)
E
Cb(E)
—
Cb(E)
We introduce the function
g(t,v)
defined
on
Let C be
[0,T]
some
x
Cb(E),
2:
Since the
g(t,n(v,t)),
with the function
k
truncating
v
in the
following
way.
positive constant, then
k(v,Ï)(x)
Step
:=
:= max
f min(C(T
All Conditions of Theorem 4.2.3
partial
(6.2.11)
differential equation
instead of Theorem 4.1.6
as
—
t),v(x)), —C(T
are
—
t) ).
fulfilled
is of first
in the additive process
order,
case
of
apply Theorem 4.2.3
Chapter 5. In fact, we only
we
will
have to show that Condition b-2 is fulfilled.
We have to prove that g is
we
a
Lipschitz
have to show that there exists
a
Cb(E), uniformly in t, i.e.
C0 independent of (t,y) E [0,T] x E such
continuous function
constant
on
that
\gy(t,vi)
-
gy(t,v2)\
<
Co\\vi
-
w2||oo
with
vi,v2 E
and
Q
=
CT.
By
the
same
C?{E)
reasoning
WL
is
Lipschitz
continuous.
:
:=
as
{v
E
Cb(E), (MU
in Section
C?{E)
-+
5.4, this
<
Q}
is ensured if
C6(supp(z/))
Proving Lipschitz continuity
goes
along
Lemma 5.3.5. Note that in the present case, all elements with
the lines of the
-k-v
dy
vanish,
and
proof of
therefore,
90
CHAPTER 6.
the derivative of
v
MEMM FOR STOCHASTIC VOLATILITY MODELS
need not be controlled.
We
now
apply
can
Theorem 4.2.3 to the
problem
ju(t,y) + 'qY^yu(t,y) + ~gy(t,ut)
u(T,y)
which
gives
Step
3:
us
a
unique weak solution
is chosen in such
k
~gy(t,ut)
u
Cb([0,T]
E
0,
(6.3.2)
=
0,
(6.3.3)
E).
x
way that
a
gy(t,ut)
=
=
We have to show that there exist
a
(t,y)E[0,T]xE.
V
constant C such that for all
(t,y)
E
[0,T]
E,
x
\u(t,y)\<(T-t)C.
For the upper
constant
C,
bound,
Then, u(s, Vfy)
—
s)C
>
for all
Therefore,
just define, for fixed (t,y)
us
[0,T]
E
x
E
as
well
as
a
positive
the deterministic time ry:
Ty
(T
let
(6.3.4)
we
s
:=
inf {s
(T-s)C
E
[t, ry),
for all
s
[t,ry)
E
and
<
(T
-
u(ry, Vff)
s)C}
<
E
s
A T.
(T-ry)C.
get (with the truncation function
we
get that, for
[t, T] | Û(s, Vfy)
E
k
Since
from step
u(s, Vfy)
1) A^Vy
>
< 0.
[t,ry),
WFt,y\K(us,s),xj
=
exp
<! Aa\yt,y'(x)
rM
r^
-
fWsM(z)WFtiy(z)u(dz)1
j Ws (x)
—^
^
[Xs
+
l
-l + XsWfW
by some constant independent of C. By the boundedness assumptions of oM,
WM and A, we then know from representation (6.3.1) that there exists a constant Ci
independent of C, such that \gy(s,us)\ < Ci for all s E [t,ry). We may write
is bounded
r-T
u(t,y)
=
/ gVs'V (s,us)ds
rT
r.Vs'V
(„
-\j„
gVs"(s,us)ds
,
+
/
I
~Vs'y
gVs"
(s,us)ds
t
gv°'y(s,Ûs)ds + Û(Ty,Vff)
<
The lower bound of
u
(Ty-t)Ci
be shown
can
Obviously, since A and WM
are
(T-ry)C.
directly.
For this purpose, let
ffWM(x)WL(x)u(dx)
2
\
consider
2
M
JWM(x)»(dx)-Xf(WM(x))2»(dx)
+
us
bounded,
1
a
+
—r—
f
/ W
M
L
(x)W (x)u(dx)
(6.3.1).
6.3.
PROOF OF THE MAIN THEOREM
and,
since in addition
aM
91
bounded,
is
-\x(cjm)2-X2 f(WM(x))2v(dx)
is also bounded.
Finally,
therefore get that
know that WL is bounded from below
we
g(s,us)
is bounded from below.
Hence, there
by
exists
XWM (x). We
1 +
—
a
constant
C2
>
0
such that
ut>-(T-t)C2.
If
fix C >
we now
Step
Let
4:
is
u
Ci
C2,
V
function g, which is
auxiliary
k is introduced in such
k(A^,t)
get (6.3.4).
differentiable in the space variable
continuously
us use an
consider
we
way that
a
instead of A"
we
slightly
different to g. A truncation function
do not bound
u
but the difference
In terms of the function g, it
.
Au,
that
means
i.e.
we
we
work
with the function
WL:Cb(E)^r(Supp(v))
defined
as
fWM(z)WL(z)u(dz)
WL(x)
e-xp\k(Au,t)(x)-
:=
X +
±
rM
I
WM(x)
M,2
M\2
(0-M)
XWM(x).
-l +
In
addition,
form with
w
to
E
g(t,ut)
that
ensure
is
differentiable,
k(w,t)(x)=< <p(w,t)(x)
{ sign(w(x))(K
some
K+
(T
fixed constants
—
uniformly
the
By
we
may
t)C,
as
ü
=
Let
+
C(T
a
g(t,ut)
=
-
following
iî\w(x)\ < (T-t)C
if (T
t)C < \w(x)\ < K
if \w(x)\ > K + (T
t)C
t))
+
(T
t)C
-
x
tp(w,t) E l°°(supp(z/)) with \tp(w,t)(x)\ <
[0,T]
l°°(su.pp(v)) is differentiable in w with
—
derivative.
Step 2,
in
Step 3,
g(t,ut).
i.e.
we
get that
provides
a
g is
Lipschitz
solution ü. Let
there exists
a
continuous and
us
consider
u
that ut
E
Since the solution must be unique,
Cl(E). By
direct
therefore,
above,
from
pair (C,K) such that k(A^,t)
we
calculation,
we
=
A"
conclude that
u.
us now assume
-
suitable
l°°(su.pp(v))
Theorem 4.2.3 which
which is bounded due to
and therefore
:
partial
reasoning
apply
that k has the
-
K and
C,
such that k
bounded
same
assume
l°°(su.pp(v)):
( v(w)
for
we
get
d
dy9y(t,ut)
^(AywKH^l^fc^fA-,)
CHAPTER 6.
92
with
Let
WL(x)
us now
WL(x)
:=
+ 1
-
MEMM FOR STOCHASTIC VOLATILITY MODELS
XWM(x)
k(t,y,Wfy(A'ly)) being uniformly
and
bounded.
write
d
-Tr9y(t,ut)
dy
dw
d
t)(x)\w=Au
k(w,
(q-&Ux)) WtLy(xMdx)
d
k(t,y,
WtLy(Aly))
d
d
—u(t,y + W^y(x))-—u(t,y))
„,v,
,.
dw
J
+
-
ju
.
d
T
/
\
—u(t,y
+k(t,y,WtLyU
°y
°y
0y-^,*,~,~~
xWfy(x)u(dx)
( fWfy(-)
„
z)dz))
+
d
=
gy(t,jr-ut).
dy
Let
us
vt(y)
set
=
-§-ut(y), which belongs to Cb(E).
continuous and bounded in
(t,y,vt)
continuous and bounded
this set. On the other
of
k,
one
directly gets
on
[0,T]
We
E
(4.2.10)
that condition
E
x
x
already
Cb(E)
know that WL is
and
hand, taking
is fulfilled
as
well
therefore,
k is
uniformly
uniformly
in account the definition
as
<9
—K(w,t)(x)
v/
W
N
w=J0
is
uniformly
are
continuous in
fulfilled and
is
6.4
The
us
E
[0, T]
the solution
therefore,
continuously differentiable
(6.2.12)
Let
(t, y, vf)
vt(y+z)dz
xExM.
Hence, all conditions of Theorem
to the PDE
u
(6.2.11)
with
boundary
4.2.4
condition
in the space variable.
Orthogonal Volatility
Process
consider the asset process
—±
=
V(t,Vt.)dt
=
rf (t,Vt-)dt
bt-
dVt
fulfilling
the
Assumptions
+
being compact
Corollary
6.4.1
d(wv( ,V-,x)*fjLY)
,
6.1.1 with
such that
The
a(t,Vt.)dYtc
+
At
and E
'"
a
is
~H
—
uniformly
optimal strategy
bounded.
Then,
we
get the following result:
is
it
=
-Xt,
(6.4.1)
THE ORTHOGONAL VOLATILITY PROCESS
6.4.
and the
whereas
density
v
is
of
process
the MEMM
=
o-L(t,Vt.)
=
of
the classical solution
+
(6.4.1)
as
given
W?(x))
+
v(t,Vt.)
0,
the
partial differential equation
vYg-v(t, y)
+
via
W>V*-
W\tVt\i,vt-,x)x)
—v(t, y)
Proof:
is
93
-
-X2a2v(t, y)
j (v(t,y Wf(x))-v(t,y)y(dx)
=
0,
(6.4.2)
v(T,y)
=
1.
(6.4.3)
+
well
as
(6.2.8). Further, (6.2.9)
aL
0 is
=
direct consequence of
a
WM(x)
=
0 and
equation
leads to
WL(t, Vt-,x)
=
exp{u(i, Vt-
+
Wf(x))
-
u(t, Vt-)}
1.
-
We know from Theorem 6.2.1 that
d
d
—u(t,y)
+
f
1-
rff—u(t,y)--Xlal
+
JWL(t,y,x)v(dx)
u(T,y)
has
a
classical solution u, which defines the MEMM.
ey+>u(t,y)
we
By using
=
0,
=
0.
v(t,y)
the transformation
=
get the linear boundary problem (6.4.2)-(6.4.3).
Remark 6.4.2
1.
specific case has already been identified by Grandits and
Rheinländer (2002) by a conditioning argument. However, while the density of the
MEMM at a fixed time T has a very simple form, the corresponding density process
turns out to be of more complicated structure.
2.
Becherer used
The
optimal strategy
jumps within
a
m
m
this
of this type
different states:
model
m
his PhD thesis.
In his case,
the process V
AC
-±
dVt
V(t,Vt-)dt
=
a(t,Vt-)dYtc
+
J2 l^t-)dNlfkf
=
3,k=l
where r\ and
indicator
a
are
function
functions of
on
This model represents
differential equation
{k}
a
class C1 with respect to t
and N
degenerate
where
ßkj
(Nkj)
case
is
a
with
multivariate
r]v
=
0 and
denotes the
adapted point
we
have the
to solve:
d
dt
=
[0,T], lfc
E
'
(t,fc)--AtVt2
v
'
2
+
J]lj/fc^(e^)-^fc)-l)
=
0
u(T,k)
=
0,
represents the jump density of
a
jump
from
state k to state j.
process.
following
CHAPTER 6.
94
MEMM FOR STOCHASTIC VOLATILITY MODELS
transformation v(t,y)
ey+>u(t,y) is here very useful since it linearizes the
partial differential equation to (6.4.2). However, this technique can not be used m
the general case when the jump process directly influences the asset process. As we
have already seen m Example 5.2.4, the exponential element cannot be linearized m
The
3.
this
=
case.
4- Benth and Meyer-Brandis (2004) determined the
simplified
generalize
BN-S model where
MEMM
for
the
special
they allow Yd to be a finite
cess
of only being a compound compensated Poisson process.
approach, the Ornstem-Uhlenbeck process a2 need not be bounded,
variation pro¬
our
results
our
result
the
m
jumps
m
that
sense
case
Brandis
a
as
the
where the parameters
vY
:=
(ß
=
—Xa2_dt
=
ß,ß,p,X
~Sff
=
(fi+ f(epx
The process a2 is
-
write
i.e. with
Corollary
apply the results
specific example
6.5.1
We
exp{
<7q
=
—
+\
>
0.
*
is
>
,
>
=
increments
0 and p <
ù(dx)dt).
only.
It
In
can
0, and ßy has
addition, Yd is
easily
be shown
^dt
+
+
at-dYtc
+
d(epx
-
1)
*
(ßy
-
At}
+
reverting towards zero and having
explicit representation of it is given by
/ exp{
Jo
—
X(t
—
with the
u(R)
assume
~
*
[V)
pos¬
u)}dYdu.
Levy
< oo
process Y
with
supp(z/)
f(e" -1M<fc)
"
=
YC +
C
E+.
Yd,
Let
whereas Yd
us
°
We denote
~
vY).
of Section 6.2. One has to pay attention to the fact that
"+
ß
(St)te\o,T\
as
the subordinator. An
a2
work in this
=
Ornstein Uhlenbeck process
an
jumps given by
We want to
dipx ßy)
+
Dy(dx,dt)
positive
a2_(ß
+
crt-dYf
real constants with A
are
l)v(dx)
+
d(x*ßy)
+
XvY (we then also
subordinator,
a
ßa2_)dt
+
that the process S may then be written
dSt
stock S
a
=
da2
assumed to be
that
sees
Model
Shephard
Shephard (2001), the price process of
exponential exp{At} with X
(Xt)t satisfying
dXt
compensator
their
the BN-S
In Barndorff-Nielsen and
by
one
m
be
m
The Barndorff-Nielsen
defined
Since
further generalized. In the following section, we will treat the
framework, including the model of Benth and Meyerspecial case but also allowing for jumps m the asset process.
might
unbounded
6.5
and
a
Their results
occur
instead
itive
of
case
the price process.
no
'
=ß
+
f(efx-l)v(dx)+ye-xt(ß + \)
ye~xt + f(ep*
l)2î>(dx)
-
assume
=
Yd.
we
THE BARNDORFF-NIELSEN SHEPHARD MODEL
6.5.
Let <7q
fixed,
0 be
>
such that
we
define
A
e
Let
us
[a^, oo).
:=
\ù(dx)
1
—
We
assume
(6.5.1)
< oo.
introduce
gy(t,ut)
\(af -Xte-^^j)2
=
WtLy(x)
where
E
95
Wf (x)
is
($t +
-
Mte-My
+
Xt)(epx
-
1)
+
ifXt(epx
-
1)21 v{dx)
defined by
exp
{ AuMx)
-
(epx
l-Xt(epx-l)
=
J(e^
1)
-
i)wfy(z)p(dz)
_
ye
+ A,
-At
WtLy(x),
+
(6.5.2)
and
A^(x)
:=
u(t,y
f(epx
2
<Pt
+
extx)-u(t,y),
l)WtLy(x)v(dx)
-
'—
ye
f(epx
[0,T]
1.
x
E1.
Then,
There exists
a
we
classical solution
dt
and
u
introduced u,
dQ
Zt
dP
Tt
Proof: The PDE
Cb([0,T]
the
boundary problem
m(0,T)xE,
u(T,y)
=
0
on
we
=
+
u.
the
us
+
follows from equation
0At
=
e
(-X(epx
-
(6.5.5)
and
aL
via
the MEMM
1)
+
WL(x))
(6.2.6) by making
=
2
at,
[ß +j(epx-l)v(dx) e-xtd2_(ß+l-))dt
+e--ixtdt-dYtc d((epx 1) (ßY îv))
+
+
-
*
-
equations
is
*
defined
(ßY
-
(6.5.2)
as
vY))
the transformation
get the dynamics
^
(6.5.4)
E.
define WL
density process of
Let
aLs)dY:
°t
we
E) of
0
~2
such that
x
=
f( f'(-Xsas
(6.5.4)
E
u(t,y)+gy(t,ut)
fix u
(6.5.3), respectively. Then,
Having
(6.5.3)
-±At
get:
d
2.
At,
l)WtLy(x)v(dx)
-
Vye
on
T
-Atw
CHAPTER 6.
96
MEMM FOR STOCHASTIC VOLATILITY MODELS
with
ex'dx
dot
Obviously, E is
apply Theorem
not
a
set and
compact
6.2.1 for
(6.5.4)-(6.5.5). Resolving
a
ßy )
*
is not bounded. We therefore may not
that there exists
proving
.
classical solution
a
this issue has turned out to be
surprisingly
u
directly
problem
to the
technical and will be
carried out in Section 6.6. One important result of Section 6.6 is that Au is bounded from
above
on
[0,T]
also aL and
Based
that
1.
u
on
(p
E.
x
are
this
Lemma
Hence, using
let
us
we
that WL and therefore
directly get
bounded.
uniformly
result,
5.3.1.1,
the Verification Procedure of Section 2.4
perform
ensuring
defines the MEMM:
Q
is
Let
an
us
equivalent probability
measure:
consider
1
U
(f)2aids
Wu(x)*ßY
+
with
Wu(x)
WL(x)
+X(epx
Since
on
E
Therefore,
1 and
0px
U has
+ 1
1)
-
WL
X(epx
-
-
1)j log (WL(x) + 1
-
-
X(epx
-
1)
WL(x).
are
locally integrable
uniformly bounded, Wu
is
uniformly
bounded.
variation and
rT
E
Hence, (2.4.2)
<
exp
is fulfilled
Wtu (x)v(dx)dt
2
the
by
< oo.
Cauchy-Schwarz inequality,
if
we can
show that
rT
E
By definition,
we
4>tot_dt
exp
have
4>t
f(epx
-xt
-
Since A is positive and WL is
bounded, (pt
E
[0,T],
(pt
<
such that for all t
On the other
hand,
4>t
since
>
Wf(x)
-xt
l)Wf(x)v(dx)
ot
introduce
a
< oo.
>
is
negative for
0 for all at-
—
f(epx
1 +
-
Xt(epx
1)(-1
1), (pt
Xt(epx
o:
=
-(/j+2)
ß
ox
big enough.
Let
us
> o.
—
+
at-
is bounded from below with
-
l))V(dx)
6.5.
THE BARNDORFF-NIELSEN SHEPHARD MODEL
97
because of
f(epx-l)v(dx) + o2_(ß+l2)
+ f(ePx
or
l)2v(dx)
ß +
A*
=
-
Let
analyze
us now
rT
E
fp2o2t_dt
exp
JO
T
E
exp<j
r-T
/
l{(Jt_<^}^crt2_dtjexp|
f
l^^^y^o^dt
Obviously,
rT
/
exp<j
is
uniformly
l{tTt_<W}<f>tat_dt
bounded. On the other
hand,
rT
E
exp<j
j
O>0t
>-(/?+J
l{at_>7f}4>tot_dt
is finite since
the set
on
{at-
Lemma 3.1,
>
0}
ensures
and condition
Q
is
a
martingale
This is
a
3.
I(Q, P)
<j (ß
exp
ai
(6.5.1), which, according
+
/
-)2
a2_dt
direct consequence of WL and iyM
to Benth et al.
(2003),
(6.5.6)
< 00.
[M, L]
are
locally
being bounded, which ensure that
By Corollary 3.2.2, Q is then a
bounded.
measure.
< 00:
We have to show that for
(epx
-
1)
*
vff
(ßY
Q-martingales. We
Integrability of the term
are
2'
measure:
the processes L and
martingale
ß
that
E
2.
1
true
-
=
vff)
WL(x)
as
well
+ 1
X(epx
-
/
as
is ensured due to the boundedness of
-
l)2
*
*zV
I (Xo
quadratic
-
aL)adt
variations
are
Q-integrable.
ùf
WL and X(epx
—
1)
second term
[adYc+ I (Xa
1)
adYc +
will show that their
(epx
-
-
aL)adt.
on
E+.
Let
us
consider the
CHAPTER 6.
98
MEMM FOR STOCHASTIC VOLATILITY MODELS
We have
adYc)T
EQ
It is well known that
=
at_dt
EQ
may write
we
rT
/
Hence,
a2_dt
fl o2_dt
Eq
4.
(6.5.1).
f -g-dS
a
By
is
Lemma
true
2.4.2,
-
showing
to
f -g-dS
exp
=
(A^l
+
x *
-
e"^-^ ßy^
*
is finite, which is, since
1/
WL is
T.
Q-martingale
E
e~XT)a2
is finite if £W
bounded, equivalent
condition
X~l(l
=
<
is
f xù(dx)
for all
true
a
Q
E
<
However, this
oo.
is fulfilled
by
the
M.e with finite relative entropy:
Q-martingale
if
show that for
we can
some
7 >
0,
0t2
772~«PJt-d[S];
7
exp^y J2at2_dt+(7?(e^-l)2*/iy)r
E
< 00.
By
(1992),
He et al.
Lemma
for 7
<
-^5-
hand,
(p
since
analogy
in
we
may
apply
is
uniformly
(6.5.6)
to
E
Hence,
exp
\ 27 /
we are
$2o2_dt
exp
u
some
constant
c
>
0,
we
get
< 00.
E
<
< 00.
by
Cauchy-Schwarz inequality
Existence of
Model,
bounded
the
7
0
In the BN-S
that
that
T
6.6
one sees
exp|(2702(e^-l)2*/iyy
E
On the other
14.39.1,
f^fd[S]t
and get
< 00.
>~>t
in the BN-S model
confronted with
a
problem
of the kind
-u(t,y)+gy(t,ut)
=
0
on[0,T)xE,
u(T,y)
=
0
on
E,
(6.6.1)
(6.6.2)
EXISTENCE OF U IN THE BN-S MODEL
6.6.
with E
[<7q, oo),
=
f(epx -l)Wf (ut,x)v(dx)
9y(t,ut)
ye
(epx
-At
ye
defined
In the
following,
compact,
u
E
will
we
Cb' ([0,T]
E).
x
-
Xzt / (epx
drop
ye-
l)2v(dx)
-
-
iyi)(dx)
the notation of
ut in
-At
+
xt
Wfy(ut,x).
+
directly apply Theorem
proceed as follows:
may not
we
/ (epx
Xt
f(epz-l)WtLy(ut,z)i>(dz)
(x)-(epx-l)
1-Xt(epx-1)
=
-
-At
as
A«
exp
l)v(dx)
/ Wfy(ut,x)v(dx)
+
Wf (ut,x)
-
A2ye-
-At
-l)Wfy(ut,x)ù(dx)
(epx
x
and
99
Wfy(x).
6.2.1 to
Obviously,
ensure
since E is not
existence of
a
solution
We will
We start with the modified
Levy
process
Yn, where the jumps
are
bounded
by
n, i.e.
l{x<n}ù. For every p E E+, we define compact sets Ep C E, increasing and with
E, on which we introduce a class of modified Barndorff-Nielsen Shephard
linip^oo Ep
vn
:=
=
models with solution up
from
some
y*
Due to this
of the
(independent
property,
we
of
can
corresponding
p) onwards, ensuring
then show that up
uff, satisfying (6.6.1)-(6.6.2)
function
PDE. We will show that
that AUp
uff
decreasing
is
is bounded from above.
converges for p
—
oo
to
a
continuous
jumps. By uniqueness of the
the corresponding classical solution vfnf By Arzela-
solution, we conclude that uff is
Ascoli, we finally prove that vf"^ then
in
case
of bounded
converges to the classical solution
u
with Au
being
bounded from above.
Step
(n)
Class of modified BN-S models with solutions up
1:
of the
corresponding
PDE
Consider
an
auxiliary
process of the
ap,o
dOptt
with ô
suitably
chosen
entiable function
as
ß'(0)
=
ß'(l)
the compact set
ß
=
E
:
0.
Ep
„2
Co;
—
V
ß
=
—
ensures
[oq,p
ne
that
ATI
o
p,t-
.At
6
(independent of p)
[0,1] with ß(z)
This
:=
type
and
=
a2t
dx *ßYn
ß being
0 for all
z
increasing, continuously differ¬
1 for all z > 1 as well
0, ß(z)
an
<
(by p
that Vp>t
=
is bounded
It is obvious
ne
=
XT
).
Let
opt- stays
us
introduce
in
Ep
for all
CHAPTER 6.
100
t E
[0,T].
Wp^y(x)
Since
MEMM FOR STOCHASTIC VOLATILITY MODELS
ß[^-jl\extx,
:=
we
corresponding gp(t,ut)
define the
:
Ep
as
f(ep*
9Ï(t,ut)
_
l)WpLt(x)vn(dx)
ye-
ye
(epx
-At
l)»n(dx)
-
/
Xt
(epx
-
l)2»n(dx)
(epx-l)WpLty(x)ùn(dx)
x
+
-
i„.„-\t
\ye
-At
/ WftJx)ùn(dx)
-
Xt
/
(epx
-
lYUdx)
with
K,t,y(x)
=
exp
{A^y(x)
-1 +
(epx
-
-
1) At
+
f(epz
~
l)WpLtJz)vn(dz)
p,t,y\
ye-
-At
At(e^-1)
and
Alt^y(x)=u[t,y + ß[^-)extx)-u(t,y).
By
Theorem 6.2.1, there exists
»
a
function up
,_
E
^1,1,
Cb' ([0,T]
x
Ep),
which is
a
classical
solution to the PDE
d_ u(t,y)+gy(t,ut)
=
0,
=
0.
dt
u(T,y)
The solution
can
be characterized
as
u(fl)(t,y)=
Let
us
all y
Step
extend the function up
to the whole set E
(6.6.3)
by setting
up
(t,y)
=
ft gp(s, 0)ds
for
> p.
2:
»
Up
is
decreasing
in y from
Using the representation (6.6.3), we
dependent of p) onwards. The proof
with
/ gl(s,ît})ds.
an
auxiliary
Lemma 6.6.1
some
y* onwards
»
will show that up
turns out to be
decreasing from
surprisingly technical.
-,,
is
some
Let
y*
us
(in¬
start
result:
The derivative ffgl(t,Ut) has the
dyiip^
—gy(t,ut)= /
form
(y—Alty(x))Wfty(x)vn(dx)
+
K(t,y,Ut)
(6.6.4)
EXISTENCE OF U IN THE BN-S MODEL
6.6.
101
with
WpLt,y(x)
as
well
WpLtJx)
:=
+ 1
-
Xt(epx
-
1)
as
!
K(t,y,ut)
(J(e^
l)WpLt(x)»n(dx))
_
=
y2e-Xt
d
dy
(epx
Xi
1) ( Wp\y(x)
-
+
(Xt
1
+
^)(epx
~
1) ) tnidx)
(6.6.5)
dy\2AtVe
Proof:
By differentiation,
—gy(t,ut)
we
/
=
+
get
-r^Wi,t,y(x)vn(dx)
\(epx
n
l)WffJx)i>n(dx)
;
-
}
>>W
ye-
n{
}
f
-xt
d
r
(er*-l)f^WpLty(x)vn(dx)
dy
+h(t,y,ut)
(6.6.6)
with
h(t,y,ut)
2Xt^-Xtj(epx-l)2Vn(dx)
dy\2AtVe
(epx
y2e-Xt
(epx
x
f(epx
-
-
l)vn(dx) -[Xt-
d
yj-Xt ) /
(epx
-
l)2vn(dx)
l)Wp\y(x)vn(dx)
-
l)Wp\y(x)vn(dx)
y2e-Xt
By
the definition of
150), Wft
Wpty
and the
Implicit
Function Theorem
(see
e.g. Zeidler
is differentiable in y with
d_
dy
d_
K,t,y(x)
dy
Kt,y(x)WpLtJx)
~K)(epx-l)[Wp\y(x)-l
(epx
-
l)Wft Jx)
P"V
y2e
(epx
-
l)WpLtJx)
ye-
By replacing
fLWpt
(x)
in
,
'
xt
-xt
equation (6.6.6),
we
(epz
(epz
-
l)Wp\y(z)vn(dz)
d
-
l)-Wp\y(z)ùn(dz).
directly get (6.6.4).
(1986),
p.
CHAPTER 6.
102
Let
us
analyze
MEMM FOR STOCHASTIC VOLATILITY MODELS
the function K in
detail. Since
some more
Xt(epx-l)<WftJx)
-l +
AUx)
< exp
(epx
-
-
l)rf(y)
-
Xt(epx
1 +
-
1)
with
)ùn(dx)
f(epx-l) (1-Xt(epx-1)
v
^
v(y)
=
L
At
,_»-xt
,
>
0,
y&
the first two terms of K
uniformly bounded on the set of decreasing functions u. On
hand,
negative for y big enough. Hence, it can be ensured
that from some y* onwards, K is negative, bounded away from zero, for any decreasing
function u(t,y + ). For the following considerations, we introduce
the other
are
the third term is
Dp:=Ep\[Q,y*).
^From equation (6.6.3)
tiation for
vp(t,y)
fj-up (t,y)
:=
-§-gp(»,upn,)
and since
T
vP(t,y)
is continuous
on
[0,T],
we
get by differen¬
that
d
9P\S) uP,sJ^s
p,
rT
K(s,y,upn})
-.Xs,
vP[s,y
+ ß(^^5
vp(s,y)
Let
us
in the
following
choose 5 >
\\ß'\\e
n
)e x)(l- ß
5
J
5
(x)ùn(dx))ds.
W
such that
,At,
1
Let
us
Since
first note that
ßf
^j1 j
=
-
/3'(^)^
vp(T,y)
0 for all y
=
0 for any y G
> p, we
negative
—vp(t,y)
with
=
Dp.
Let
E
us
[0,T].
now
consider the
case
y > p.
get
vP(t,y)=
We have to show that vp is
for all t
> 0
on
/
K(s,y,0)ds<0.
[0,T]
Xi(t,y,4rat)
x
+
(y*,p).
For this purpose, let
vp(t,y)K2(t,y,upn})
us
write
(6.6.7)
EXISTENCE OF U IN THE BN-S MODEL
6.6.
Ki(t,y,upn})
103
-K(t,y,upnl)
=
,At,
vP(t,y /3(^)eAtx)(l-/rr(^)^)l^tiI,(a;)i/n(d2:),
+
K2(t,y,upnf)
One
=
Since up
-ejf K^v4f)ds f Ki^ ^ q(n)y- jj K2(r,y,û^)drds_
=
is continuous
[0,T]
E
Wp^y(x)vn(dx).
represent vp in the following way:
can
Vp(t, y)
t.
I
as
well
up(T,y)
as
[0,t.),
In this
there exists
situation,
9
a
on
there exists
Dp,
some
=
0,
vp(t0,y)
<
0,
•)
+
[t.,T]
E
this set.
Let
us
x
(6.6.8)
Dp.
now
that for
assume
some
such that
vP(t0,yo)
(to,yo
up
negative
Dp
yo G
(t,y)
for all
0
>
We therefore conclude that vp is
G
0 for all y G
=
such that
Kx(t,y,vgl)
to
vp(T,y)
=
for all y
and
decreasing,
is
>
y0.
therefore,
,
-7^vp(to,y0)
-K(t0,yo,upnl)
3Ato^
-y^(to,yo /3(^)e^)(l-/3'(^)^)^0),0(x)^^)
+
>0.
In
a
small
neighborhood
is contained in
(yo,p)-
of to,
taking
vP(ti,yi)
=
0,
vp(ti,y)
<
0
continuation that there must be
[0,T)
x
Step
3:
We
now
Dp. Hence,
we
{(t, y)
Since the set
stands in contradiction with
any t\ >
some
(6.6.8).
E
y. G
of
have to show that
continuous function. For
for all y
[0, T]
(yo,p)
>
conclude that yi
(t,y)
yi
0}
Dp \ vp(t, y)
such that vp(t.,y.)
x
is
fulfilling
=
a
decreasing
x
Dp
is
=
closed,
get by
However, this
0.
we
strictly negative
function
on
[0,T]
x
on
Dp.
(up )p
(up )p
given
By representation (6.6.3),
we can
We therefore conclude that vp is
have shown that up
Convergence
to,
up
converges
q E
is
E+,
we
on
[0, T]
towards
a
show uniform convergence
negative
and
uniformly
decreasing, bounded,
on
the set
bounded from
[0, T]
below,
x
Dq.
hence
CHAPTER 6.
104
If
equibounded.
we
can
MEMM FOR STOCHASTIC VOLATILITY MODELS
show that the sequence
(uP )p
Arzela-Ascoli that there exists
this,
f^fup
will show that
we
is
vp
To do
QyUp
=
—
equicontinuous
subsequence
a
bounded function which is
Vp
directly get by
[0,T] x Dq to a continuous
we
is
on
{uff )
[0,T]
x
converging
Dq,
on
decreasing.
uniformly
bounded from below.
Let
us
consider
K\
for
u(t,y
functions
inf
fixed,
0
>
some e
=
+
\K(t,y,ut)
|
which is finite since
). Assuming
p > q, we
vP(t,y)
For any t
[0,T) fixed,
E
-QjVp{t,y)
=
E\[a%,y*),u(t,y*
y G
we
=
is
) decreasing
uniformly
j
-
e
bounded for
decreasing
introduce
vp(t,y)
vp(t,p)
have
K(t,y,ut)
+
>
-
j
Kids.
0 and
Kl-K(t,y,upnl)
+
T
jf
r
Kids
„,x
„At,
J ß'{^)e-^W^y(x)Vn(dx)
'i)P(t,y /3(^)e^)(l-^(^)^)^>)i/n(d:r),
+
+vp(t,y)K2(t,y,upnl).
This
partial
differential equation has the identical structure
Kl-K(t,y)
the
steps
Kids
above,
as
j' ß>(l^y^WpLtJx)vn(dx)
we can
show that vp is
<0.
strictly positive
on
[0, T)
x
Dp,
andtherefore,
same
J
however
andtherefore,
Performing
+
(6.6.7),
as
i
on
[0,T]
Hence
x
we
vp is bounded from below. We
can
conclude that
(up )p>q
is
equicontinuous
Dq.
get
u
(n)
(t,y)
=
Mm upn)(t,y)
p—>oo
"p
rT
=
um
p—>oo
/ 9p(s,upn)s)ds
't
T
lim gp(s,upn's)ds
p\") "'p.s/"'"
p^oo
,
,-T
lim
p—>oo
't
gy(s,upn's)ds
T
gy(s,
lim upnl)ds
p^°°
t
p
T
j gy(s,u^s)ds.
(since
\"
vv
gp(-,upn)) is
ap\
1
p.-
bounded)
6.6.
EXISTENCE OF U IN THE BN-S MODEL
Therefore, uff
Levy
process
However, the
is the solution
uf1^
to the
boundary problem (6.6.1)-(6.6.2)
with modified
Yn.
sequence
(;u(-"'-))ra
by Arzela-Ascoli
is decreasing from
is
equibounded
and equicontinuous
therefore
converges to the classical solution
since
y* onwards.
u
105
u.
on
compact sets, and
Au is bounded from above
106
CHAPTER 6.
MEMM FOR STOCHASTIC VOLATILITY MODELS
A
Appendix
Levy
Processes
This section
gives
short overview of
a
lot of interest in the last few years.
divisible distribution
infinitely
of time At.
On the other
Levy
On the
one
be chosen
can
hand, they
Levy
hand, they
processes.
have
as
a
simple
(Vl,T,P)
Let
They
flexible,
two
the
a
since any
over
periods
comparison with
gen¬
Levy
Lévy-Itô Decomposition.
representations
on
Lévy-Khintchine Representation
be
probability space. Levy
following way:
a
processes
are
a
subclass of
a
Levy
semimartingales.
may be defined in the
conditions
are
Xo
2.
Independent
0
in
Ed
is
t2
<
process
if
the
following
satisfied:
1.
=
(Xt)t>o
A stochastic process
Definition A. 1.1
a.s.
increments:
of
Xto, Xt2
For any choice
Xto, Xtl
3.
very
structure in
semimartingales. In the following, we concentrate on
processes, namely the Lévy-Khintchine Representation and
The
are
the increment distribution
eral
A.l
processes have attracted
-
Stationary
>
n
-
1
and 0 < to
Xtl,..., Xtn
-
<
Xtn_1
t\
<
are
...
tn, the random variables
<
independent.
increments:
The distribution
4- Stochastically
For every t >
of Xs+t
—
Xs does
not
depend
on s.
continuous:
0,
e
>
0
limP[|Xs-Xt| >e]
=
0.
s—*t
5.
There
is
Qo
continuous
We define
5. A
Levy
an
m
E
T with
P[Qo]
left
t > 0 and has
additive process
process in law is
a
as a
=
1
such
limits
m
that, for
uj
E
satisfying
is
the conditions 1, 2,
process when the condition 5 is
107
VLq, Xt(oj)
right-
t > 0.
stochastic process
Levy
every
dropped.
4, and
APPENDIX A.
108
A suitable way to
purpose, let
us
classify Levy
processes is the
introduce the class of
Definition A. 1.2
Consider
a
ß(z)
The law ß
is
infinitely
called
In other
words,
Example
from
1.
their
A.1.3
be
following distributions
characteristic function:
Compound
concentrated at
3.
c >
4- (d
0 and
some
distribution
=
1)
a
d
x
d
tl(z)
=
7 G
=
a
Ed with
on
(d
=
1)
=
probability
which
can
be
seen
Ed
1)}
—
o{0}
exp{-- (z, Az)
T-distribution with parameter
=
=
0.
where q
its
c >
+
1
0 and
—
a >
=
1
—
p.
The
p(l
-
negative binomial distribution with parameter
divisible distributions
distribution
1
qe'T1
ß(z)=pc(l-qen-c-
Uniform
0
< p <
generalization:
infinitely
Ed.
ia~lz)}.
Geometric distribution with parameter p, 0
=
(7, z)}
matrix and 7 G
exp{—clog(l
ß(z)
on
are
following:
the
[—a, a]
ß(z)
2.
a
el{l'z).
exp{c(a(z)
=
positive-definite symmetric
ß(z)
1.
is
Gaussian Distribution
with A
Not
function
Poisson Distribution
ß(z)
5.
its characteristic
infinitely divisible,
are
single point
a
ß(z)
for
Ed and
=
The
5-distribution,
on
ß
exp{« (z,x)}ß(dx).
ß(z)
2.
measure
if for any positive integer n, there
ßff1.
function ß~n such that ß
expressed as the n-th convolution power of ßn.
ßn with characteristic
can
/
=
For this
divisible distributions.
divisible
measure
ß
Lévy-Khintchine Representation.
infinitely
probability
LEVY PROCESSES
(sin az) / (az).
=
Binominal distribution with parameters n,p
ß(z)
=
(pelx
+
q)n.
c >
0 and p
is
A.l.
In
fact,
The
LÉVY-KHINTCHINE
THE
probability
no
(other
measure
following theorem gives
distributions, which
6-y)
than
109
with bounded support
infinitely
is
divisible.
representation of the characteristic function of infinitely
a
divisible
REPRESENTATION
is called the
Lévy-Khintchine representation:
Theorem A. 1.4
1.
If ß
is
infinitely
an
divisible distribution
Rd,
on
then
ß(z)
(AAA)
a
=
measure
| ^ (z,Az)
{x
\\x\\
on
-
:
1},
<
A
If
A
ing
was
is
/
Al)u(dx)
is
v
is
(A.l.2)
< oo,
unique.
symmetric, nonnegative-definite d
a
(A. 1.2)
in the
7)
v,
and 7
G
Ed;
then there exists
function
obtained in E around 1930
by Levy
symmetric nonnegative-definite dx d matrix,
a
0 and
=
by (A,
whose characteristic
It
(el{z'x) -l-i{z,x) lD(x))i/(cte)},
d
G
The representation
2.
is
j
+
Ed satisfying
u({0})
and 7
i{j,z)
+
UFII2
where D
exp
(\\x\\2
=
general
given
unique
v
infinitely
is
a
measure
satisfy¬
divisible distribution ß
by (A.1.1).
by de Finetti and Kolmogorov in special cases and
proof in Ed, consult Sato (1999), Theorem 8.1.
For
case.
is
a
d matrix,
x
then
a
Remark A. 1.5
1.
(A,u,j)
We call
Gaussian
2.
The
coefficient
integral of
respect
to
Theorem
m
the
and the
measure
integrand
because it
v
is
A.l.4
generating triplet of
the
v
is
called the
such
The
z.
right-hand side of (A.1.1)
outside of any neighborhood of
bounded
a
ß(z)
with 70
=
7
(A,u,jo)0.
/11
n —^-,
J\\x\\>l
—
=
(A,v, 71)1
-
L
=
exp
7 +
00,
'
\
L
-
then
-
2
(z, Az)
-
is
we
is
+
1
(70, z)
This may be
here called the
is
integrable
with
0 and
0
+
I (et{z'x)
l)v(dx)
-
JRd
the
represented by
drift of ß. If
v
generating triplet
satisfies
the condition
qet
3
(z, Az)
f,,,,>lxu(dx).
and 71
\x\->
as
of ß.
still
2
The constant 70
ß(z)
=
{
0(\\x\\2)
called the
the
on
fVd xlo(x)u(dx).
||:r||z/(<ir) <
II
II
V
/
with 71
f\\x\\<x
exp
=
is
right-hand side of (A.1.1) may be altered m
is
integrable for every z. Thus, if v satisfies the
\x\v(dx) < oo; then, using the zero function, we get
function Id
integrand
way that the
additional condition
measure
the
m
e^z'x>-l-i{z,x)lD(x)
for fixed
Levy
A
ß.
+
1
(71, z)
+
This will be
called the center.
/ (e1^'^
-
1
-
JRd
represented by
1
(z, x))v(dx)
the
generating triplet
Let
LEVY PROCESSES
APPENDIX A.
110
us now
mention
ible distributions and
which shows the
theorem,
a
Levy
correspondence
between
divis¬
infinitely
processes.
Theorem A. 1.6
1.
If (Xt)t>o is a Levy process on Rd, then, for any t > 0, the distribution of Xt
is infinitely divisible and, letting Pxx
ßl', i-e.
ß, we have Pxt
=
-
E[et{u'Xt)]
-
Pxt
=
=
exp{tV(«)}
with
rtp(u)
2.
--(u,Au)+i(-f,u)+
=
Conversely, if ß
process
(Xt)t>o
is
an
infinitely
such that
Proofs of this theorem
can
Pxx
=
\ (et{u'x)
-
l-i(u,x)lD(x))u(dx)
divisible distribution
ß- It
is
on
unique up to
be found in Sato
(1999),
Rd,
then there exists
identity
Theorem
(A.l.3)
m
7.10,
Theorem 1.1 and the comments prior to it. The first statement is
a
Levy
law.
or
Bertoin
directly
seen
(1996),
from the
structure of
Levy processes. The second statement is proven in two steps. First, we
construct a Levy process in law using Kolmogorov's extension theorem. Second, it can be
proven that every Levy process in law has a modification which is a Levy process. This
goes back to Doob. The triplet (A, v, 7) is called the characteristics of X.
A.2
The
Lévy-Itô Decomposition
Another way to characterize
studying the sample paths. The
Lévy-Itô decomposition expresses sample functions of a Levy process as a sum of two
independent parts a continuous part and a part expressible as a compensated sum of
independent jumps. We will later explain what we mean by a compensated sum. The
decomposition in a continuous part and a jump part was conceived by Levy and formulated
and proved by Itô (1942). Let us first introduce the following definition:
Levy
processes goes via
-
Definition A.2.1
{ß(B) : B
{0,1, 2,..., +oo};
ables
(0,B,p) be a a-fimte measure space. A family of random vari¬
B}, defined on a probability space (Vl,T,P) and taking values m
Let
E
is
called
B, ß(B)
a
Poisson random
1.
for
2.
for any n and for
independent,
any
3.
for
ß(B,uj),
Now,
we
every
every
fixed
uj,
formulate the
measure
has Poisson distribution with
disjoint B\, ...,Bn,
B E
B,
is
intensity
mean
measure
p
if
p(B),
the random variables
a measure.
Lévy-Itô decomposition:
with
ß(Bi), ...,ß(Bn)
are
A.2.
LÉVY-ITÔ
THE
Theorem A.2.2
(Vl,T,P)
(Xt)t>o
Let
1.1)
A.
ßx((0,t\,G,u)
ßx(dt,dx)
u(dx)
and
we
be
Levy
a
111
is
,
and
uj
probability
space
Qo (H0
using
E
#{sE(0,t];Xs(u)-Xs_(u)EG} foruEÜo,
foru^üo^
0
X0(uj)
=
B^d
E
a
f
.-
may write that there
Xt(uj)
For any G
on
let
Poisson random
a
Ed defined
on
process
generating triplet (A,u,y).
with
from Definition
Then
DECOMPOSITION
Xt(uj)
+
Vl\
is
with
measure
+ lim
e^°
Pfàf\
T with
E
/
J{e<||ii;||<l}
xßx((0,t], dx,oj)
+
intensity
=
ux(dt,dx)
measure
for
1 such that
{xßx((0, t], dx,uj)
any
dt
=
E
uj
x
Vl\,
xtu(dx)}
—
Ff
\\x\\>l}
X0(uj)+XÏ(uj)+Xd(uj)+t"f,
=
where the convergence
is
m
process with
Levy
a
t
uniform
m
t
on
any bounded time interval.
generating triplet (0,
v,
generating triplet (A, 0,0).
with
a.s.
is
(A.2.1)
0)
(X^)t>o
and
is
a
Levy
process continuous
(Xd)t>o
The two processes
(Xf)t>o
The process
and
(X^)t>o
are
independent.
The
proof
can
be found in Sato
(1999),
Theorem 19.2.
Remark A.2.3
1.
One
can
show that
as e
I
is
0
of jumps
2.
Levy
The
ance
fr<,,,,<1,{xßx((0,t],dx) xtu(dx)}
called the
compensated
fr,,x,,<1yX[j,x((Q,t\)dx)
process
matrix
A)
has
—
Xt
and
is
a
of jumps.
Without the
may not converge
linear combination
quadratic
a
sum
of
a
=
as e
0.
Hence,
subtraction,
[ £(An)2)c
Yt,
=
its limit
the
sum
10.
Brownian motion Bt
pure jump process
iyyt
mean
(with
covari-
i.e.
o
0<s<t
for
any t E
E+.
In
Yt(u)
addition,
=
Xf(u)
+
tE^Yx
-
f
J{\\*\\>n
xv(dx)]
=
Xf(u)
+
yt,
with
y
the
3.
drift of
the
Levy
Let
us now
classify
the
E\Vi-
/
xu(dx)],
process.
To shorten the notation,
Xd(oj)
=
=
Levy
we
write
xlD(x)
*
measures:
(ßX
-
vx)
+
(x- xlD(x))
*
ßX-
(A.2.2)
APPENDIX A.
112
{Xt}
be
said to be
of
Definition A.2.4 Let
Levy
measure v
type A
ifu(Rd)
type B if
type C
is
u(Rd)
a
Lxh<1 \\x\\u(dx)
oo; and
=
2/J|M|<i ||:r||z/(<ir)
=
Sample functions of (Xt)t>o
2.
If u(Rd)
m
a
<
3.
for
u(Rd)
v
proof
;
then,
is
< oo;
00.
order,
that
v
Such
is
continuous
are
a.s.
if
and
only if
surely, jump times are countable
process of infinite activity.
a
then,
almost
and the
a
surely, jump times
first jump time T(uj)
is
process
of type
A
called
B.
or
a
Then,
process
a.s.,
u
=
0.
and dense
m
[0,oo).
infinitely many and countable
has exponential distribution with
of finite activity.
are
Xf(uj)
has
finite
variation
on
(0,t\
(0, 00).
of type C, then,
on (0, t] for any t
is
Ed with generating triplet (A,u,y). A
almost
called
< oo;
l/v(Rd).
any t E
tion
The
00
process
Suppose
If
=
increasing
mean
on
< oo;
1.
IfO
process
A.2.5
Properties
Such
Levy
LEVY PROCESSES
almost
E
of these properties
surely,
the
sample function
Xf(uj)
has
infinite
varia¬
(0, 00).
can
be found in Sato
treated in Theorems 21.1, 21.3 and 21.9,
respectively.
(1999),
with properties 1, 2 and 3
Index of Notation
Chapter
2:
definition
:=
(Ü,T,P)
probability space
finite, fixed time horizon
T
F
=
(Ft)o<t<T
filtration
p
objective probability
E[-]
Eq[-]
expectation with respect
to P
expectation with respect
to
(St)o<t<T
HQ,P)
(discounted)
relative entropy of
M
set of all
S
=
probability
measure
Q
asset process
Q
w.r.t. P
feds
martingale measures
set of equivalent martingale measures
set of martingale measures with finite relative entropy
set of S'-inegrable predictable processes
set of portfolio strategies
trading strategies
stochastic integral: gains from trade
®M
set of
Me
Mf
L(S)
0 C
L(S)
9,4>eO
L(S)
such that
for all
Q EMf
u
utility
function
u
maximal
QB
Q°
ns(B)
MB)
n°(e)
L
.
•—
dQ
dP
(Zt)o<t<T
'=
y^p
£()
ANt
is
a
(<5,F)-martingale
expected utility function
random variable: contingent claim
optimal martingale measure
minimal entropy martingale measure (MEMM)
utility indifference (selling) price
utility indifference hedging strategy
utility indifference price for the contingent claim
B
ry
f 9dS
)
density
of
density
process
Q
^t/0<t<T
Doléans-Dade
:=
Nt
-
Nt-
of size eB
w.r.t. P
jump
(stochastic) exponential
function
size of process N at time t
•A-loc
set of all
adapted processes with locally integrable
of Aioc, containing increasing processes
A+
subset
113
variation
INDEX OF NOTATION
114
Chapter
3
Y
Levy
yc
continuous
ßy
jump
Vy
compensator of the jump
VQ
process
of Y
martingale part
measure
of Y
measure
Vy
compensator of ßy under Q
M
local
Mc,Md
[X,Y]
(X,Y)
[X]
(X)
Qiocißy)
LL(YC)
continuous and discontinuous
martingale
martingale part
predictable covariation process of X and Y (if it exists)
quadratic variation process of X
predictable quadratic variation process of X (if it exists)
set of predictable functions W on Q x [0,T] x E (see Not. 3.1.1)
set of all predictable processes H such that f H2d(Yc)
locally integrable
4
differential equation
PDE
partial
A
differential operator
(C6r(R),||HI|r)
(cm, m -ni)
normed space of
Banach space of Holder-continuous functions
continuous
lll^lll
l,r
•—
bounded,
derivative, equipped
11^11°°
4.1.1)
bounded,
\\^ ll°°
'
set of
bounded, continuous
bounded, continuous space
functions
C6o,1([0,T) xE)nC1'2((0,T) xE)
fundamental solution of
min (a,
Ab
E
an
parabolic
[0,T] x E with
on [0,T) x E
on
derivatives
r
PDE
b)
set in E
Chapter
H(ST)
=
5
h(logST)
European contingent claim
u(t,-) :E^E
u(t,y + f2tt(x)) -u(t,y)
ut
c
C<
supp(z/)
Z°°(supp(i/))
K
Def.
with supremums-norm
C*([0,T)xE)
C*>L
(see
continuous functions with
c61(E)nr(E)
C°b\[0,T)xR)
a
of M
covariation process of X and Y
is
Chapter
ßy
set of functions
support of
Levy
v
E
Cb'r(R)
measure
set of bounded functions
truncation function
stopping
time
v
with
<
Q
and
\w'(y)\
< L
115
Chapter
6
BN-S model
Barndorff-Nielsen
Wv
jump heights of the volatility
ßy
jump
Yn
modified
measure
Levy
Shephard
of Yd
=
model
process
Yf
process with
jumps bounded by
n
116
INDEX OF NOTATION
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BIBLIOGRAPHY
Curriculum Vitae
Personal Data
Name:
Date of Birth:
Gallus
Steiger
September 15,
Nationality:
Swiss
Civil Status:
married
1972
School
1978-1984
1984-1992
Primary School, Meggen
Kantonsschule Alpenquai,
Matura
Type
Luzern
C
Undergraduate
Studies
1992-1994
Studies in Mathematics at
1994-1995
Studies in Mathematics at Freie Universität and Technische
University
of
Fribourg
Universität, Berlin
1995-1997
Studies in Mathematics at Universität of
1997
Diploma
Diploma
Fribourg
in Mathematics
thesis under the supervision of Prof. F.
Delbaen,
ETH Zürich
Ph.D. Studies
2001-2005
Ph.D. student at ETH Zürich
Supervisors: Prof. T. Rheinländer & Prof.
Topic: Hedging and Pricing in Incomplete
F. Delbaen
Markets
Professional Data
1997-2003
Ernst &
2003-2005
ETH Zürich
2005
Teaching Assistant, Researcher in the NCCR FINRISK
Swiss Re, Global Asset and Risk Management, Zürich
Senior
Young,
Manager
Risk
Management Services, Zürich,
Vice President
123
New York
program