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Bernt ØksendalStochastic Differential EquationsAn Introduction with ApplicationsFifth Edition, Corrected PrintingSpringer-Verlag Heidelberg New YorkSpringer-VerlagBerlin Heidelberg NewYorkLondon Paris TokyoHong Kong BarcelonaBudapest

To My FamilyEva, Elise, Anders and Karina

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The front cover shows four sample paths Xt (ω1 ), Xt (ω2 ), Xt (ω3 ) and Xt (ω4 )of a geometric Brownian motion Xt (ω), i.e. of the solution of a (1-dimensional)stochastic differential equation of the formdXt (r α · Wt )Xtdtt 0 ; X0 xwhere x, r and α are constants and Wt Wt (ω) is white noise. This process isoften 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 setof new questions. In some ways we feel we are as confusedas ever, but we believe we are confused on a higher leveland about more important things.Posted outside the mathematics reading room,Tromsø University

Preface to Corrected Printing, Fifth EditionThe main corrections and improvements in this corrected printing are fromChaper 12. I have benefitted from useful comments from a number of people, including (in alphabetical order) Fredrik Dahl, Simone Deparis, UlrichHaussmann, Yaozhong Hu, Marianne Huebner, Carl Peter Kirkebø, Nikolay Kolev, Takashi Kumagai, Shlomo Levental, Geir Magnussen, AndersØksendal, Jürgen Potthoff, Colin Rowat, Stig Sandnes, Lones Smith, Setsuo Taniguchi and Bjørn Thunestvedt.I want to thank them all for helping me making the book better. I alsowant to thank Dina Haraldsson for proficient typing.Blindern, May Bernt Øksendal

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Preface to the Fifth EditionThe 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 toinclude this material as another major application of stochastic analysis, inview of the amazing development in this field during the last 10–20 years.Moreover, the close contact between the theoretical achievements and theapplications in this area is striking. For example, today very few firms (ifany) trade with options without consulting the Black & Scholes formula!The first 11 chapters of the book are not much changed from the previousedition, but I have continued my efforts to improve the presentation throughout and correct errors and misprints. Some new exercises have been added.Moreover, to facilitate the use of the book each chapter has been dividedinto subsections. If one doesn’t want (or doesn’t have time) to cover all thechapters, then one can compose a course by choosing subsections from thechapters. The chart below indicates what material depends on which sections.Chapter 1-5Chapter 8Section8.6Chapter 7Chapter 10Chapter 12Chapter 6Section9.1Chapter 9Chapter 11Section12.3For example, to cover the first two sections of the new chapter 12 it is recommended that one (at least) covers Chapters 1–5, Chapter 7 and Section 8.6.

VIIIChapter 10, and hence Section 9.1, are necessary additional background forSection 12.3, in particular for the subsection on American options.In my work on this edition I have benefitted from useful suggestionsfrom many people, including (in alphabetical order) Knut Aase, Luis Alvarez, Peter Christensen, Kian Esteghamat, Nils Christian Framstad, HelgeHolden, Christian Irgens, Saul Jacka, Naoto Kunitomo and his group, SureMataramvura, Trond Myhre, Anders Øksendal, Nils Øvrelid, Walter Schachermayer, Bjarne Schielderop, Atle Seierstad, Jan Ubøe, Gjermund Våge andDan Zes. I thank them all for their contributions to the improvement of thebook.Again Dina Haraldsson demonstrated her impressive skills in typing themanuscript – and in finding her way in the LATEX jungle! I am very gratefulfor her help and for her patience with me and all my revisions, new versionsand revised revisions . . .Blindern, January 1998Bernt Øksendal

Preface to the Fourth EditionIn this edition I have added some material which is particularly useful for theapplications, namely the martingale representation theorem (Chapter IV),the variational inequalities associated to optimal stopping problems (ChapterX) and stochastic control with terminal conditions (Chapter XI). In additionsolutions and extra hints to some of the exercises are now included. Moreover,the proof and the discussion of the Girsanov theorem have been changed inorder to make it more easy to apply, e.g. in economics. And the presentationin general has been corrected and revised throughout the text, in order tomake the book better and more useful.During this work I have benefitted from valuable comments from severalpersons, including Knut Aase, Sigmund Berntsen, Mark H. A. Davis, HelgeHolden, Yaozhong Hu, Tom Lindstrøm, Trygve Nilsen, Paulo Ruffino, IsaacSaias, Clint Scovel, Jan Ubøe, Suleyman Ustunel, Qinghua Zhang, TushengZhang and Victor Daniel Zurkowski. I am grateful to them all for their help.My special thanks go to Håkon Nyhus, who carefully read large portionsof the manuscript and gave me a long list of improvements, as well as manyother useful suggestions.Finally I wish to express my gratitude to Tove Møller and Dina Haraldsson, who typed the manuscript with impressive proficiency.Oslo, June 1995Bernt Øksendal

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Preface to the Third EditionThe main new feature of the third edition is that exercises have been includedto each of the chapters II–XI. The purpose of these exercises is to help thereader to get a better understanding of the text. Some of the exercises arequite routine, intended to illustrate the results, while other exercises areharder and more challenging and some serve to extend the theory.I have also continued the effort to correct misprints and errors and toimprove the presentation. I have benefitted from valuable comments andsuggestions from Mark H. A. Davis, Håkon Gjessing, Torgny Lindvall andHåkon Nyhus, My best thanks to them all.A quite noticeable non-mathematical improvement is that the book isnow typed in TE X. Tove Lieberg did a great typing job (as usual) and I amvery grateful to her for her effort and infinite patience.Oslo, June 1991Bernt Øksendal

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Preface to the Second EditionIn the second edition I have split the chapter on diffusion processes in two, thenew Chapters VII and VIII: Chapter VII treats only those basic propertiesof diffusions that are needed for the applications in the last 3 chapters. Thereaders that are anxious to get to the applications as soon as possible cantherefore 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 arecentral in today’s theory of stochastic analysis and crucial for many otherapplications.Hopefully this change will make the book more flexible for the differentpurposes. I have also made an effort to improve the presentation at somepoints and I have corrected the misprints and errors that I knew about,hopefully without introducing new ones. I am grateful for the responses thatI have received on the book and in particular I wish to thank Henrik Martensfor his helpful comments.Tove Lieberg has impressed me with her unique combination of typingaccuracy and speed. I wish to thank her for her help and patience, togetherwith Dina Haraldsson and Tone Rasmussen who sometimes assisted on thetyping.Oslo, August 1989Bernt Øksendal

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Preface to the First EditionThese notes are based on a postgraduate course I gave on stochastic differential equations at Edinburgh University in the spring 1982. No previousknowledge about the subject was assumed, but the presentation is based onsome background in measure theory.There are several reasons why one should learn more about stochasticdifferential equations: They have a wide range of applications outside mathematics, there are many fruitful connections to other mathematical disciplinesand the subject has a rapidly developing life of its own as a fascinating research field with many interesting unanswered questions.Unfortunately most of the literature about stochastic differential equations seems to place so much emphasis on rigor and completeness that itscares many nonexperts away. These notes are an attempt to approach thesubject from the nonexpert point of view: Not knowing anything (except rumours, maybe) about a subject to start with, what would I like to know firstof all? My answer would be:1) In what situations does the subject arise?2) What are its essential features?3) 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 ratherin an easier proof of a special case, which may give just as much of the basicidea in the argument. And I would be willing to believe some basic resultswithout proof (at first stage, anyway) in order to have time for some morebasic applications.These notes reflect this point of view. Such an approach enables us toreach the highlights of the theory quicker and easier. Thus it is hoped thatthese notes may contribute to fill a gap in the existing literature. The courseis meant to be an appetizer. If it succeeds in awaking further interest, thereader will have a large selection of excellent literature available for the studyof the whole story. Some of this literature is listed at the back.In the introduction we state 6 problems where stochastic differential equations play an essential role in the solution. In Chapter II we introduce thebasic mathematical notions needed for the mathematical model of some ofthese problems, leading to the concept of Ito integrals in Chapter III. InChapter IV we develop the stochastic calculus (the Ito formula) and in Chap-

XVIter V we use this to solve some stochastic differential equations, including thefirst two problems in the introduction. In Chapter VI we present a solutionof the linear filtering problem (of which problem 3 is an example), usingthe stochastic calculus. Problem 4 is the Dirichlet problem. Although this ispurely deterministic we outline in Chapters VII and VIII how the introduction of an associated Ito diffusion (i.e. solution of a stochastic differentialequation) leads to a simple, intuitive and useful stochastic solution, which isthe cornerstone of stochastic potential theory. Problem 5 is an optimal stopping problem. In Chapter IX we represent the state of a game at time t by anIto diffusion and solve the corresponding optimal stopping problem. The solution involves potential theoretic notions, such as the generalized harmonicextension provided by the solution of the Dirichlet problem in Chapter VIII.Problem 6 is a stochastic version of F.P. Ramsey’s classical control problemfrom 1928. In Chapter X we formulate the general stochastic control problem in terms of stochastic differential equations, and we apply the results ofChapters VII and VIII to show that the problem can be reduced to solvingthe (deterministic) Hamilton-Jacobi-Bellman equation. As an illustration wesolve a problem about optimal portfolio selection.After the course was first given in Edinburgh in 1982, revised and expanded versions were presented at Agder College, Kristiansand and University of Oslo. Every time about half of the audience have come from the applied section, the others being so-called “pure” mathematicians. This fruitfulcombination has created a broad variety of valuable comments, for which Iam 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. andNorges Almenvitenskapelige Forskningsråd (NAVF), Norway for their financial support. And I am greatly indebted to Ingrid Skram, Agder College andInger Prestbakken, University of Oslo for their excellent typing – and theirpatience with the innumerable changes in the manuscript during these twoyears.Oslo, June 1985Bernt ØksendalNote: Chapters VIII, IX, X of the First Edition have become Chapters IX,X, XI of the Second Edition.

Table of Contents1.Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.1 Stochastic Analogs of Classical Differential Equations . . . . . . .1.2 Filtering Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.3 Stochastic Approach to Deterministic Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.4 Optimal Stopping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.5 Stochastic Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.6 Mathematical Finance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11223442.Some Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . 72.1 Probability Spaces, Random Variables and Stochastic Processes 72.2 An Important Example: Brownian Motion . . . . . . . . . . . . . . . . . 11Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.Itô Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.1 Construction of the Itô Integral . . . . . . . . . . . . . . . . . . . . . . . . . .3.2 Some properties of the Itô integral . . . . . . . . . . . . . . . . . . . . . . . .3.3 Extensions of the Itô integral . . . . . . . . . . . . . . . . . . . . . . . . . . . .Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21213034374.The Itô Formula and the Martingale Representation Theorem . .