Springer-Verlag Berlin Heidelberg GmbH

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1 Universitext

2 Springer-Verlag Berlin Heidelberg GmbH

3 Peter E. Kloeden Henri Schurz Eckhard Platen Numerical Solution of SDE Through Computer Experiments With 55 Figures Springer

4 Peter E. Kloeden Fachbereich Mathematik Johann Wolfgang Goethe-Universitiit Frankfurt am Main, Germany Eckhard Platen Department of Mathematical Sciences University of Technology Broadway, NSW 2007, Australia Henri Schurz Department of Mathematics Southern Illinois University Carbondale, IL , USA Corrected Third Printing 2003 Cataloging-in-Publication Data applied for Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at< Mathematics Subject Classification (2000): 60HlO, 93E30, 93Ell, 90A09, 93D05 ISBN ISBN (ebook) DOI / This work is subject to copyright. AII rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg 1994 Originally published by Springer-Verlag Berlin Heidelberg New York in 1994 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: design & production GmbH, Heidelberg Typeset by the authors Printed on acid-free paper SPIN /3111LK

5 Contents Preface... VII Legal Matters... XI Introduction... XIII Chapter 1: Background on Probability and Statistics Probability and Distributions Random Number Generators Moments and Conditional Expectations Random Sequences Testing Random Numbers Markov Chains as Basic Stochastic Processes Wiener Processes Chapter 2: Stochastic Differential Equations Stochastic Integration Stochastic Differential Equations Stochastic Taylor Expansions... '" Chapter 3: Introduction to Discrete Time Approximation Numerical Methods for Ordinary Differential Equations A Stochastic Discrete Time Simulation Pathwise Approximation and Strong Convergence Approximation of Moments and Weak Convergence Numerical Stability..., Chapter 4: Strong Approximations Strong Taylor Schemes Explicit Strong Schemes Implicit Strong Approximations Simulation Studies Chapter 5: Weak Approximations Weak Taylor Schemes Explicit Weak Schemes and Extrapolation Methods Implicit Weak Approximations

6 VI CONTENTS 5.4 Simulation Studies Variance Reducing Approximations Chapter 6: Applications Visualization of Stochastic Dynamics Testing Parametric Estimators Filtering Functional Integrals and Invariant Measures Stochastic Stability and Bifurcation Simulation in Finance References Subject Index List of PC-Exercises Frequently Used Notations

7 Preface The numerical solution of stochastic differential equations is becoming an indispensible worktool in a multitude of disciplines, bridging a long-standing gap between the well advanced theory of stochastic differential equations and its application to specific examples. This has been made possible by the much greater accessibility to high-powered computers at low-cost combined with the availability of new, effective higher order numerical schemes for stochastic differential equations. Many hitherto intractable problems can now be tackled successfully and more realistic modelling with stochastic differential equations undertaken. The aim of this book is to provide a computationally oriented introduction to the numerical solution of stochastic differential equations, using computer experiments to develop in the readers an ability to undertake numerical studies of stochastic differential equations that arise in their own disciplines and an understanding, intuitive at least, of the necessary theoretical background. It is related to, but can also be used independently of the monograph P.E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Applications of Mathematics Series Vol. 23, Springer-Verlag, Heidelberg, 1992, which is more theoretical, presenting a systematic treatment of time-discretized numerical schemes for stochastic differential equations along with background material on probability and stochastic calculus. To facilitate the parallel use of both books, the presentation of material in this book follows that in the monograph closely. The results of over one hundred personal computer exercises were used in the monograph to illustrate many of the concepts and theoretical matters under discussion. An immediate objective of the present book is to provide the solutions for these PC-Exercises in the form of Borland TURBO PASCAL programs on a floppy diskette together with commentaries on the structure of the programs and hints on their use. The book is directed at beginners to stochastic differential equations and their numerical solution. Readers will typically be upper level undergraduates or graduates in diverse disciplines with several semesters background in deterministic calculus and basic mathematical methods together with an exposure to probability and statistics. A familiarity with elementary programming and the ability to interact with IBM compatible PCs using MS-DOS and Borland TURBO PASCAL are also assumed (see the end of the preface for other languages and other computing environments). The emphasis is quite practical

8 VIII PREFACE and mathematical proofs are avoided, but, as with all meaningful numerical work, an understanding of the underlying theoretical issues is as important for the numerical solution of stochastic differential equations as are hands-on computational skills. While non-mathematicians amongst the readers may not aspire to a mastery of the stochastic calculus on which the numerical schemes for stochastic differential equations are based, they should be aware that it has some peculiar features which are quite different from those of the familiar deterministic calculus and may lead to serious errors if disregarded. Readers are urged to delve more deeply into these important theoretical matters if they have plans for specific, nontrivial numerical investigations. The organisation and contents of the book are based on courses that the co-authors have taught at the Australian National University, the Humboldt University in Berlin, the University of Hamburg and the University of Technology in Sydney. The PC-Exercises from the above monograph, which are called Problems here, form the backbone of the book. Background material on probability, random number generation, stochastic processes and statistics are reviewed in Chapter 1 by means of a large number of these Problems. The programs, in particular the key computational and graphics subroutines that reappear repeatedly in the sequel, are explained in some detail in this chapter, which should be read if only for this reason. Chapter 2 provides a descriptive introduction to stochastic calculus and stochastic differential equations, including stochastic Taylor expansions and their constituent multiple stochastic integrals from which the higher order numerical schemes for stochastic differential equations are derived. The numerical solution of initial value problems for differential equations via time-discretized numerical schemes is surveyed in Chapter 3, first for deterministic ordinary differential equations and then for stochastic differential equations. Computing exercises for differential equations with known explicit solutions are used to illustrate theoretical convergence rates and the effects of round-off error and numerical instability. For stochastic differential equations a fundamental distinction is made between strong, pathwise approximations and weak, distributional approximations. Different requirements for each are discussed in the context of the Euler scheme applied to a simple linear stochastic differential equation. Chapters 4 and 5 are devoted to higher order schemes for strong and weak approximations, respectively. Comparative numerical studies are presented there, along with many practical hints based on the experience of the co-authors and their co-workers over the past 15 years. The final chapter, Chapter 6, introduces the reader to the vast range of situations in which the numerical simulation of stochastic differential equations can playa useful role, including stability and bifurcation in stochastic dynamical systems, filtering, testing parametric estimators, calculating invariant measures, and finance modelling. In each case several applied simulation projects are formulated as Exercises for the reader. The diskette contains Borland TURBO PASCAL programs for all of the Problems in Chapters 1 to 5. They are based on programs and subroutines prepared by the co-authors and their co-workers, who are not professional programmers, over the past five years and are made available here solely for the

9 PREFACE IX convenience of the readers, to spare them the tedium of preparing their own programs and to enable them to adapt or modify the programs to their own problems. The PASCAL language was chosen because of its universality and to highlight the transparency of the structure and the adaptability of the programs, which usually took priority over computational efficiency in their preparation. Preliminary instructions on their use are given in the 'README' file on the diskette. The programs typically require at most a few minutes running time on a 386 PC with numerics processor, though some of those in the later chapters may need an hour or two to give acceptable accuracy in terms of small confidence intervals. In most cases the equations and parameters in the programs can be easily changed. Readers are encouraged to do so in a copy of the original program and to experiment as much as possible. The authors would like to thank Katrin Platen who word-processed the manuscript and Steven Dutch, David Heath, John van der Hoek, Norbert Hofmann, Ivan Huerta, Igor Shkurko, Robert Stine and Raymond Webster who developed some of the programs, tested many of the others, and carefully read the manuscript. In addition they would like to thank Ludwig Arnold, Daryl Daley and Chris Heyde and their respective institutions for support and encouragement. Canberra, June 1993 Peter E. Kloeden Eckhard Platen Henri Schurz Authors' Comments on the Reprinting We are delighted by the overwhelming response to this book from readers in all areas of applications, in particular in finance, that has led to its being sold out within a few years of its first appearance at the end of This reprinting has provided us with the opportunity to correct some minor misprints and to update the status of some papers previously listed in the References as to appear. Unfortunately other commitments have prevented us from developing versions of the software in other programming languages as we had hoped to do and had mentioned in a now deleted paragraph in the original version of the above Preface. We thank all readers for their helpful feedback and welcome further comments from readers. May 1997

10 x PREFACE Authors' Comments on the Third Printing Earlier printings of this book included a diskette containing the TURBO PAS CAL programs that are used. The rapid developments of the Internet in recent years now make it both cheaper and more convenient to make these programs available on each of our homepages staff/ eckhard.html Readers can also find other useful software and information about stochastic differential equations, their applications and the methods for their numerical solution on these homepages. Note that wherever "diskette" is used in the book, it should be interpreted as "downloadable software". There are too many occurrences of this, naturally, too numerous to make a correction of each and everyone possible in practical term. We hope that this will not be too bothersome for the reader. August 2002

11 Legal Matters We make no warranties, expressed or implied, that the programs contained in this book and the downloadable software are free of error, or are consistent with any particular standard of merchantability, or that they will meet your requirements for any particular application. They should not be relied on for solving a problem that could result in injury to a person or loss of property. If you do use the programs in such a manner, it is at your own risk. The authors and publisher disclaim all liability for direct or consequential damages resulting from your use of the programs. Several registered trademarks appear in the book: IBM is a trademark of International Business Machines Corporation, MS-DOS and Microsoft C are trademarks of Microsoft Corporation, THINK PASCAL is a trademark of Symantec Corporation, TURBO PASCAL and TURBO C are trademarks of Borland International, Inc., UNIX is a trademark of AT&T Bell Laboratories, Macintosh is a trademark of Apple Computing, Inc. and MATHEMATICA is a trademark of Wolfram Research, Inc. Computer programs can be protected by copyright like artistic and literary compositions. Generally it is an infringement for you to copy into your computer a program from a copyrighted source. Although this book and its programs and their future versions in other programming languages are copyrighted, we specifically authorize you, a reader of the book, to make one machine-readable copy of the programs for your own use. We have been careful and have tried to avoid mistakes, but if you find any errors or bugs, please document them and tell us. The downloadable software requires the user to have installed Borland TURBO PASCAL 5.0 or later versions. Comments on how to run the programs can be found in the 'README' file.

12 Introd uction The theory of stochastic differential equations (SDEs) has been in existence for nearly fifty years now. Complicated finite dimensional stochastic dynamics can now be modelled and understood theoretically through the Ito stochastic calculus and the more general theory of semimartingales. SDEs now find application as a matter of course in diverse disciplines such as in civil and mechanical engineering, economics and finance, environmental science, signal processing and filtering, chemistry and physics, population dynamics and psychology, pharmacology and medicine, to mention just a few. Their role in modelling continuous time stochastic dynamics is comparable to that of deterministic ordinary differential equations (ODEs) in nonrandom differentiable dynamics. In some applications, such as to stock market prices, the deterministic component in the dynamics is often only of secondary interest. Other fields, such as random vibrations in mechanical systems, require stochastic models more sophisticated than earlier, purely deterministic ones for an adequate description of the noise effects. Very few specific SDEs have explicitly known solutions, these being mainly reducible by appropriate transformation to the solution of linear SDEs. The computation of important characteristics such as moments or sample paths for a given SDE is thus crucial for the effective practical application of SDEs. This requires methods that are specific not only to SDEs but also for the desired task. The visualization of sample paths of SDEs, as in stochastic flows or phase portraits of noisy oscillators can be achieved by use of quite natural discrete time strong approximation methods. These provide algorithms in which the SDE is discretized in time and approximations to its solution at the discretization instants are computed recursively, with values at intermediate times being obtained through interpolation. An approximate sample path of the SDE is thus obtained for a given sample path of the driving Wiener process, which is usually simulated with an appropriately chosen pseudo random number generator. In filtering the observation process is playing the role of the driving Wiener process in the Zakai equation. Higher order methods can be derived from stochastic Taylor expansions which involves multiple stochastic integrals of the driving Wiener process in the SDEs. Such schemes are, however, only useful when they are stable numerically, an issue that attracts much attention. Many applications require the computation of a functional of a solution of an SDE such as a moment or the expectation of a terminal pay-off. Numerically this is easier because only a probability measure of the solution has to

13 XIV INTRODUCTION be approximated rather than its highly irregular sample paths, but there are nevertheless still many practical complications. Higher order schemes for this weak kind of convergence can also be constructed systematically from stochastic Taylor expansions, but now the driving noise integrals can be approximated by simpler, more easily generated random variables depending on the desired order of the scheme. Other numerical methods are also available for the computation of such functionals, which are often the solution of a partial differential equation (PDE) related to the Kolmogorov backward or Fokker-Planck equation. In principle, standard finite-difference or finite-element methods could be used, but these require both spatial and temporal discretization and their complexity increases exponentially in the spatial dimension. Such an approach remains intractable even with present day supercomputers for most higher dimensional SDEs. A more probabilistic method based on Markov chains acting at discrete time instants on a discretized state space has been developed, but this is also limited in higher dimensions and it is difficult to derive higher order schemes and to undertake a stability analysis. In principle, the binomial and trinomial tree methods in finance modelling have much in common with this Markov chain approach. The classical Monte-Carlo methods may seem more attractive in comparsion since their complexity increases only polynomially with the spatial dimension, but accuracy of statistical estimates of the simulated functionals is still only inversely proportional to the square root of the number of samples used and a huge number of sample paths may be required when large variances are involved. Higher order methods and a stability analysis also do not fit well within the framework of such Monte-Carlo methods even though variance reduction techniques are available. The emphasis in this book will focus on strong and weak numerical schemes that are based on discrete time approximations of SDEs. Unlike the PDE and Markov chain approaches, they involve the whole state space since only the time variable is discretized. While the complexity of weak schemes also increases polynomially in the state space dimension like that of the Monte-Carlo methods, the direct use of the specific structure of the given SDE greatly facilitates the derivation of more efficient higher order schemes and numerically stable algorithms. This approach allows to use much from the internal structur of the problem at hand by exploiting specific properties and tools based on stochastic analytic ideas as martingales and corresponding measure transformations. Thus modern stochastic analysis allows maximal use of the information in a given SDE to be incorporated in such weak time discretized methods, in particular with control over variances considerably superior to those of Monte Carlo methods.