Bernt Øksendal
Stochastic Differential Equations
An Introduction with Applications
Fifth Edition, Corrected Printing
Springer-Verlag Heidelberg New York
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Stochastic Differential Equations, Exercises of Differential Equations
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Bernt Øksendal
Stochastic Differential Equations
An Introduction with Applications
Fifth Edition, Corrected Printing
Springer-Verlag Heidelberg New York
Springer-Verlag
Berlin Heidelberg NewYork
London Paris Tokyo
Hong Kong Barcelona
Budapest
To My Family
Eva, Elise, Anders and Karina
The front cover shows four sample paths Xt(ω 1 ), Xt(ω 2 ), Xt(ω 3 ) and Xt(ω 4 ) of a geometric Brownian motion Xt(ω), i.e. of the solution of a (1-dimensional) stochastic differential equation of the form
dXt dt
= (r + α · Wt)Xt t ≥ 0 ; X 0 = x
where x, r and α are constants and Wt = Wt(ω) is white noise. This process is often used to model “exponential growth under uncertainty”. See Chapters 5, 10, 11 and 12. The figure is a computer simulation for the case x = r = 1, α = 0.6. The mean value of Xt, E[Xt] = exp(t), is also drawn. Courtesy of Jan Ubøe, Stord/Haugesund College.
We have not succeeded in answering all our problems. The answers we have found only serve to raise a whole set of new questions. In some ways we feel we are as confused as ever, but we believe we are confused on a higher level and about more important things.
Posted outside the mathematics reading room, Tromsø University
VI
Preface to the Fifth Edition
The main new feature of the fifth edition is the addition of a new chapter, Chapter 12, on applications to mathematical finance. I found it natural to include this material as another major application of stochastic analysis, in view of the amazing development in this field during the last 10–20 years. Moreover, the close contact between the theoretical achievements and the applications in this area is striking. For example, today very few firms (if any) trade with options without consulting the Black & Scholes formula! The first 11 chapters of the book are not much changed from the previous edition, but I have continued my efforts to improve the presentation through- out and correct errors and misprints. Some new exercises have been added. Moreover, to facilitate the use of the book each chapter has been divided into subsections. If one doesn’t want (or doesn’t have time) to cover all the chapters, then one can compose a course by choosing subsections from the chapters. The chart below indicates what material depends on which sections.
Chapter 8
Chapter 1-
Chapter 7
Chapter 10
Chapter 6
Chapter 9
Chapter 11
Section Chapter 12 12.
Section
Section
For example, to cover the first two sections of the new chapter 12 it is recom- mended that one (at least) covers Chapters 1–5, Chapter 7 and Section 8.6.
Preface to the Fourth Edition
In this edition I have added some material which is particularly useful for the applications, namely the martingale representation theorem (Chapter IV), the variational inequalities associated to optimal stopping problems (Chapter X) and stochastic control with terminal conditions (Chapter XI). In addition solutions and extra hints to some of the exercises are now included. Moreover, the proof and the discussion of the Girsanov theorem have been changed in order to make it more easy to apply, e.g. in economics. And the presentation in general has been corrected and revised throughout the text, in order to make the book better and more useful. During this work I have benefitted from valuable comments from several persons, including Knut Aase, Sigmund Berntsen, Mark H. A. Davis, Helge Holden, Yaozhong Hu, Tom Lindstrøm, Trygve Nilsen, Paulo Ruffino, Isaac Saias, Clint Scovel, Jan Ubøe, Suleyman Ustunel, Qinghua Zhang, Tusheng Zhang and Victor Daniel Zurkowski. I am grateful to them all for their help. My special thanks go to H˚akon Nyhus, who carefully read large portions of the manuscript and gave me a long list of improvements, as well as many other useful suggestions. Finally I wish to express my gratitude to Tove Møller and Dina Haralds- son, who typed the manuscript with impressive proficiency.
Oslo, June 1995 Bernt Øksendal
X
XII
Preface to the Second Edition
In the second edition I have split the chapter on diffusion processes in two, the new Chapters VII and VIII: Chapter VII treats only those basic properties of diffusions that are needed for the applications in the last 3 chapters. The readers that are anxious to get to the applications as soon as possible can therefore jump directly from Chapter VII to Chapters IX, X and XI. In Chapter VIII other important properties of diffusions are discussed. While not strictly necessary for the rest of the book, these properties are central in today’s theory of stochastic analysis and crucial for many other applications. Hopefully this change will make the book more flexible for the different purposes. I have also made an effort to improve the presentation at some points and I have corrected the misprints and errors that I knew about, hopefully without introducing new ones. I am grateful for the responses that I have received on the book and in particular I wish to thank Henrik Martens for his helpful comments. Tove Lieberg has impressed me with her unique combination of typing accuracy and speed. I wish to thank her for her help and patience, together with Dina Haraldsson and Tone Rasmussen who sometimes assisted on the typing.
Oslo, August 1989 Bernt Øksendal
Preface to the First Edition
These notes are based on a postgraduate course I gave on stochastic dif- ferential equations at Edinburgh University in the spring 1982. No previous knowledge about the subject was assumed, but the presentation is based on some background in measure theory. There are several reasons why one should learn more about stochastic differential equations: They have a wide range of applications outside mathe- matics, there are many fruitful connections to other mathematical disciplines and the subject has a rapidly developing life of its own as a fascinating re- search field with many interesting unanswered questions. Unfortunately most of the literature about stochastic differential equa- tions seems to place so much emphasis on rigor and completeness that it scares many nonexperts away. These notes are an attempt to approach the subject from the nonexpert point of view: Not knowing anything (except ru- mours, maybe) about a subject to start with, what would I like to know first of all? My answer would be:
- In what situations does the subject arise?
- What are its essential features?
- What are the applications and the connections to other fields?
I would not be so interested in the proof of the most general case, but rather in an easier proof of a special case, which may give just as much of the basic idea in the argument. And I would be willing to believe some basic results without proof (at first stage, anyway) in order to have time for some more basic applications. These notes reflect this point of view. Such an approach enables us to reach the highlights of the theory quicker and easier. Thus it is hoped that these notes may contribute to fill a gap in the existing literature. The course is meant to be an appetizer. If it succeeds in awaking further interest, the reader will have a large selection of excellent literature available for the study of the whole story. Some of this literature is listed at the back. In the introduction we state 6 problems where stochastic differential equa- tions play an essential role in the solution. In Chapter II we introduce the basic mathematical notions needed for the mathematical model of some of these problems, leading to the concept of Ito integrals in Chapter III. In Chapter IV we develop the stochastic calculus (the Ito formula) and in Chap-
XVI
ter V we use this to solve some stochastic differential equations, including the first two problems in the introduction. In Chapter VI we present a solution of the linear filtering problem (of which problem 3 is an example), using the stochastic calculus. Problem 4 is the Dirichlet problem. Although this is purely deterministic we outline in Chapters VII and VIII how the introduc- tion of an associated Ito diffusion (i.e. solution of a stochastic differential equation) leads to a simple, intuitive and useful stochastic solution, which is the cornerstone of stochastic potential theory. Problem 5 is an optimal stop- ping problem. In Chapter IX we represent the state of a game at time t by an Ito diffusion and solve the corresponding optimal stopping problem. The so- lution involves potential theoretic notions, such as the generalized harmonic extension provided by the solution of the Dirichlet problem in Chapter VIII. Problem 6 is a stochastic version of F.P. Ramsey’s classical control problem from 1928. In Chapter X we formulate the general stochastic control prob- lem in terms of stochastic differential equations, and we apply the results of Chapters VII and VIII to show that the problem can be reduced to solving the (deterministic) Hamilton-Jacobi-Bellman equation. As an illustration we solve a problem about optimal portfolio selection. After the course was first given in Edinburgh in 1982, revised and ex- panded versions were presented at Agder College, Kristiansand and Univer- sity of Oslo. Every time about half of the audience have come from the ap- plied section, the others being so-called “pure” mathematicians. This fruitful combination has created a broad variety of valuable comments, for which I am very grateful. I particularly wish to express my gratitude to K.K. Aase, L. Csink and A.M. Davie for many useful discussions. I wish to thank the Science and Engineering Research Council, U.K. and Norges Almenvitenskapelige Forskningsr˚ad (NAVF), Norway for their finan- cial support. And I am greatly indebted to Ingrid Skram, Agder College and Inger Prestbakken, University of Oslo for their excellent typing – and their patience with the innumerable changes in the manuscript during these two years.
Oslo, June 1985 Bernt Øksendal
Note: Chapters VIII, IX, X of the First Edition have become Chapters IX, X, XI of the Second Edition.
- Introduction
- 1.1 Stochastic Analogs of Classical Differential Equations
- 1.2 Filtering Problems
- lems 1.3 Stochastic Approach to Deterministic Boundary Value Prob-
- 1.4 Optimal Stopping
- 1.5 Stochastic Control
- 1.6 Mathematical Finance
- Some Mathematical Preliminaries
- 2.1 Probability Spaces, Random Variables and Stochastic Processes
- 2.2 An Important Example: Brownian Motion
- Exercises
- Itˆo Integrals
- 3.1 Construction of the Itˆo Integral
- 3.2 Some properties of the Itˆo integral
- 3.3 Extensions of the Itˆo integral
- Exercises
- rem 4. The Itˆo Formula and the Martingale Representation Theo-
- 4.1 The 1-dimensional Itˆo formula
- 4.2 The Multi-dimensional Itˆo Formula
- 4.3 The Martingale Representation Theorem
- Exercises
- Stochastic Differential Equations
- 5.1 Examples and Some Solution Methods
- 5.2 An Existence and Uniqueness Result
- 5.3 Weak and Strong Solutions
- Exercises
- The Filtering Problem XVIII Table of Contents
- 6.1 Introduction
- 6.2 The 1-Dimensional Linear Filtering Problem
- 6.3 The Multidimensional Linear Filtering Problem
- Exercises
- Diffusions: Basic Properties
- 7.1 The Markov Property
- 7.2 The Strong Markov Property
- 7.3 The Generator of an Itˆo Diffusion
- 7.4 The Dynkin Formula
- 7.5 The Characteristic Operator
- Exercises
- Other Topics in Diffusion Theory
- 8.1 Kolmogorov’s Backward Equation. The Resolvent
- 8.2 The Feynman-Kac Formula. Killing
- 8.3 The Martingale Problem
- 8.4 When is an Itˆo Process a Diffusion?
- 8.5 Random Time Change
- 8.6 The Girsanov Theorem
- Exercises
- Applications to Boundary Value Problems
- 9.1 The Combined Dirichlet-Poisson Problem. Uniqueness
- 9.2 The Dirichlet Problem. Regular Points
- 9.3 The Poisson Problem
- Exercises
- Application to Optimal Stopping
- 10.1 The Time-Homogeneous Case
- 10.2 The Time-Inhomogeneous Case
- 10.3 Optimal Stopping Problems Involving an Integral
- 10.4 Connection with Variational Inequalities
- Exercises
- Application to Stochastic Control
- 11.1 Statement of the Problem
- 11.2 The Hamilton-Jacobi-Bellman Equation
- 11.3 Stochastic control problems with terminal conditions
- Exercises
- Application to Mathematical Finance Table of Contents XIX
- 12.1 Market, portfolio and arbitrage
- 12.2 Attainability and Completeness
- 12.3 Option Pricing
- Exercises
- Appendix A: Normal Random Variables
- Appendix B: Conditional Expectation
- gence Appendix C: Uniform Integrability and Martingale Conver-
- Appendix D: An Approximation Result
- Solutions and Additional Hints to Some of the Exercises
- References
- List of Frequently Used Notation and Symbols
- Index