Applied stochastic control of jump diffusions

214 Pages·2007·1.294 MB·English

by  Bernt Øksendal, Agnès Sulem

·
Bernt Øksendal Agnès Sulem
Applied Stochastic
Control
of Jump Diffusions
With24Figures
123
BerntØksendal
UniversityofOslo
CenterofMathematicsforApplications(CMA)
DepartmentofMathematics
0316Oslo
Norway
e-mail:[email protected]
AgnèsSulem
INRIARocquencourt
DomainedeVoluceau
78153LeChesnayCedex
France
e-mail:[email protected]
Mathematics Subject Classification (2000): 93E20, 60G40, 60G51, 49L25,
65MXX,47J20,49J40,91B28
Coverfigureistakenfrompage91.
LibraryofCongressControlNumber:2004114982
ISBN3-540-14023-9SpringerBerlinHeidelbergNewYork
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To my family
Eva, Elise, Anders and Karina
B. Ø.
A tous ceux qui m’accompagnent
A. S.
Preface
Jump diffusions are solutions of stochastic differential equations driven by
L´evyprocesses.Since a L´evyprocessη(t) canbe written asa linear combina-
tion of t, a Brownian motion B(t) and a pure jump process, jump diffusions
represent a natural and useful generalization of Itoˆ diffusions. They have re-
ceived a lot of attention in the last years because of their many applications,
particularly in economics.
There exist today several excellent monographs on L´evy processes. How-
ever,veryfew ofthem - ifany - discussthe optimalcontrol,optimalstopping
andimpulsecontrolofthecorrespondingjumpdiffusions,whichisthesubject
of this book. Moreover, our presentation differs from these books in that it
emphazises the applied aspect of the theory. Therefore we focus mostly on
useful verification theorems and we illustrate the use of the theory by giving
examplesandexercisesthroughoutthe text.Detailedsolutionsofsomeofthe
exercises are given in the end of the book. The exercices to which a solution
is provided, are marked with an asterix ∗. It is our hope that this book will
fill a gap in the literature and that it will be a useful text for students, re-
searchers and practitioners in stochastic analysis and its many applications.
Although most of our results are motivated by examples in economics and
finance, the results are general and can be applied in a wide variety of sit-
uations. To emphasize this, we have also included examples in biology and
physics/engineering.
This book is partially based on courses given at the Norwegian School of
EconomicsandBusinessAdministration(NHH)inBergen,Norway,duringthe
Spring semesters 2000 and 2002, at INSEA in Rabat, Morocco in September
2000,atOdenseUniversityinAugust2001andatENSAEinParisinFebruary
2002.
VIII Preface
Acknowledgments Wearegratefultomanypeoplewhoinvariouswayshavecon-
tributedtotheselecturenotes.InparticularwethankKnutAase,FredEspenBenth,
Jean-Philippe Chancelier, Rama Cont, Hans Marius Eikseth, Nils Christian Fram-
stad, Jørgen Haug, Monique Jeanblanc, Kenneth Karlsen, Thilo Meyer-Brandis,
Cloud Makasu, Sure Mataramvura, Peter Tankov and Jan Ubøe for their valuable
help. We also thank Francesca Biagini for useful comments and suggestions to the
textandherdetailedsolutionsofsomeoftheexercises.WearegratefultoDinaHar-
aldsson andMartine Verneuillefor proficienttypingand EivindBrodal forhiskind
assistance. We acknowledge with gratitude the support by the French-Norwegian
cooperation project Stochastic Control and Applications, Aur99–050.
Oslo and Paris, August 2004 Bernt Øksendal and Agn`es Sulem
Contents
1 Stochastic Calculus with Jump diffusions .................. 1
1.1 Basic definitions and results on L´evy Processes ............. 1
1.2 The Itoˆ formula and related results ........................ 5
1.3 L´evy stochastic differential equations ...................... 10
1.4 The Girsanov theorem and applications .................... 12
1.5 Application to finance.................................... 19
1.6 Exercises ............................................... 22
2 Optimal Stopping of Jump Diffusions...................... 27
2.1 A general formulation and a verification theorem ............ 27
2.2 Applications and examples................................ 31
2.3 Exercises ............................................... 36
3 Stochastic Control of Jump Diffusions .................... 39
3.1 Dynamic programming................................... 39
3.2 The maximum principle .................................. 46
3.3 Application to finance.................................... 52
3.4 Exercises ............................................... 55
4 Combined Optimal Stopping and Stochastic Control
of Jump Diffusions......................................... 59
4.1 Introduction ............................................ 59
4.2 A general mathematical formulation ....................... 60
4.3 Applications ............................................ 65
4.4 Exercises ............................................... 69
5 Singular Control for Jump Diffusions ..................... 71
5.1 An illustrating example .................................. 71
5.2 A general formulation.................................... 73
5.3 Application to portfolio optimization with transaction costs... 76
5.4 Exercises ............................................... 78
X Contents
6 Impulse Control of Jump Diffusions ....................... 81
6.1 A general formulation and a verification theorem ............ 81
6.2 Examples............................................... 85
6.3 Exercices............................................... 94
7 Approximating Impulse Control of Diffusions
by Iterated Optimal Stopping.............................. 97
7.1 Iterative scheme......................................... 97
7.2 Examples...............................................107
7.3 Exercices...............................................112
8 Combined Stochastic Control and Impulse Control
of Jump Diffusions.........................................113
8.1 A verification theorem ...................................113
8.2 Examples...............................................116
8.3 Iterative methods........................................120
8.4 Exercices...............................................122
9 Viscosity Solutions ........................................123
9.1 Viscosity solutions of variational inequalities ................124
9.2 The value function is not always C1 ........................127
9.3 Viscosity solutions of HJBQVI ............................130
9.4 Numerical analysis of HJBQVI............................140
9.5 Exercises ...............................................146
10 Solutions of Selected Exercises.............................149
10.1 Exercises of Chapter 1 ...................................149
10.2 Exercises of Chapter 2 ...................................153
10.3 Exercises of Chapter 3 ...................................162
10.4 Exercises of Chapter 4 ...................................169
10.5 Exercises of Chapter 5 ...................................171
10.6 Exercises of Chapter 6 ...................................174
10.7 Exercises of Chapter 7 ...................................185
10.8 Exercises of Chapter 8 ...................................188
10.9 Exercises of Chapter 9 ...................................191
References.....................................................197
Notation and Symbols .........................................203
Index..........................................................207
1
Stochastic Calculus with Jump diffusions
1.1 Basic definitions and results on L´evy Processes
Inthischapterwepresentthebasicconceptsandresultsneededfortheapplied
calculus ofjump diffusions. Since there areseveralexcellent bookswhich give
a detailed account of this basic theory, we will just briefly review it here and
refer the reader to these books for more information.
Definition 1.1.Let (Ω,F,{Ft}t≥0,P) be a filtered probability space. An Ft-
adapted process {η(t)}t≥0 = {ηt}t≥0 ⊂ R with η0 = 0 a.s. is called a L´evy
process if ηt is continuous in probability and has stationary, independent in-
crements.
Theorem 1.2.Let{ηt}beaL´evyprocess.Thenηt hasacadlagversion(right
continuous with left limits) which is also a L´evy process.
Proof. See e.g. [P], [S]. (cid:3)(cid:4)
Inview of this resultwe will from nowonassume that the L´evyprocesses
we work with are cadlag.
The jump of ηt at t≥0 is defined by
∆ηt =ηt−ηt− . (1.1.1)
Let B0 be the family of Borel sets U ⊂ R whose closure U¯ does not contain
0. For U ∈B0 we define
(cid:1)
N(t,U)=N(t,U,ω)= XU(∆ηs). (1.1.2)
s:0<s≤t
In other words, N(t,U) is the number of jumps of size ∆ηs ∈U which occur
before or at time t. N(t,U) is called the Poisson random measure (or jump
measure) of η(·). The differential form of this measure is written N(dt,dz).
2 1 Stochastic Calculus with Jump diffusions
Remark 1.3.Note that N(t,U) is finite for all U ∈ B0. To see this we
proceed as follows: Define
T1(ω)=inf{t>0;ηt ∈U}
We claim that T1(ω) > 0 a.s. To prove this note that by right continuity of
paths we have
lim η(t)=η(0)=0 a.s.
t→0+
Therefore, for all ε > 0 there exists t(ε) > 0 such that |η(t)| < ε for all
t<t(ε). This implies that η(t)(cid:7)∈U for all t<t(ε), if ε<dist(0,U).
Next define inductively
Tn+1(ω)=inf{t>Tn(ω);∆ηt ∈U}.
Then by the above argument Tn+1 >Tn a.s. We claim that
Tn →∞ as n→∞, a.s.
Assume not. Then Tn →T <∞. But then
lim η(s) cannot exist.
s→T−
contradicting the existence of left limits of the paths.
Itiswell-knownthatBrownianmotion{B(t)}t≥0 hasstationaryandinde-
pendentincrements.ThusB(t)isaL´evyprocess.Anotherimportantexample
is the following:
Example 1.4(ThePoissonprocess).ThePoissonprocessπ(t)ofintensity
λ>0 is a L´evy process taking values in N∪{0} and such that
(λt)n
P[π(t)=n]= e−λt ; n=0,1,2,...
n!
Theorem 1.5.[P, Theorem 1.35].
(i) The set function U →N(t,U,ω)defines a σ-finitemeasureon B0 for each
fixed t,ω.
(ii) The set function
ν(U)=E[N(1,U)] (1.1.3)
where E = EP denotes expectation with respect to P, also defines a σ-finite
measure on B0, called the L´evy measure of {ηt}.
(iii) Fix U ∈B0. Then the process
πU(t):=πU(t,ω):=N(t,U,ω)
is a Poisson process of intensity λ=ν(U).