Modeling with Itoˆ Stochastic Differential Equations MATHEMATICAL MODELLING: Theory and Applications VOLUME 22 Thisseriesisaimedatpublishingworkdealingwiththedefinition,development and application of fundamental theory and methodology, computational and algorithmic implementations and comprehensive empirical studies in mathe- matical modelling. Work on new mathematics inspired by the construction of mathematical models, combining theory and experiment and furthering the understanding of the systems being modelled are particularly welcomed. 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Laubenbacher, (Virginia Bioinformatics Institute, Virginia Tech, USA) The titles published in this series are listed at the end of this volume. Modeling with Itoˆ Stochastic Differential Equations By E. Allen Texas Tech University, USA AC.I.P.CataloguerecordforthisbookisavailablefromtheLibraryofCongress. ISBN-13978-1-4020-5952-0(HB) ISBN-13978-1-4020-5953-7(e-book) PublishedbySpringer, P.O.Box17,3300AADordrecht,TheNetherlands. www.springer.com Printed on acid-free paper AllRightsReserved (cid:1)c 2007Springer Nopartofthisworkmaybereproduced,storedinaretrievalsystem,ortransmittedin anyformorbyanymeans,electronic,mechanical,photocopying,microfilming,recording orotherwise,withoutwrittenpermissionfromthePublisher,withtheexceptionofany materialsuppliedspecificallyforthepurposeofbeingenteredandexecutedonacomputer system,forexclusiveusebythepurchaserofthework. To Linda and Anna Contents Preface ........................................................ xi 1 Random Variables ......................................... 1 1.1 Introduction ............................................ 1 1.2 Probability Space........................................ 2 1.3 Random Variable, Probability Distribution ................. 4 1.4 Expectation ............................................ 7 1.5 Multiple Random Variables ............................... 9 1.6 A Hilbert Space of Random Variables ...................... 13 1.7 Convergence of Sequences of Random Variables ............. 17 1.8 Computer Generation of Random Numbers ................. 20 1.9 Monte Carlo ............................................ 23 Exercises ................................................... 27 Computer Programs.......................................... 30 2 Stochastic Processes ....................................... 33 2.1 Introduction ............................................ 33 2.2 Discrete Stochastic Processes ............................. 34 2.3 Continuous Stochastic Processes........................... 39 2.4 A Hilbert Space of Stochastic Processes .................... 45 2.5 Computer Generation of Stochastic Processes ............... 50 2.6 Examples of Stochastic Processes.......................... 52 Exercises ................................................... 55 Computer Programs.......................................... 59 3 Stochastic Integration ..................................... 63 3.1 Introduction ........(cid:1).................................... 63 t 3.2 Integrals of the Form f(s,ω)ds ......................... 63 a 3.3 Itˆo Stochastic Integrals................................... 67 3.4 Approximation of Stochastic Integrals...................... 72 3.5 Stochastic Differentials and Itˆo’s Formula................... 74 vii viii Contents 3.6 Stratonovich Stochastic Integrals .......................... 80 3.7 Multidimensional Itoˆ’s Formula ........................... 82 Exercises ................................................... 83 Computer Programs.......................................... 87 4 Stochastic Differential Equations .......................... 89 4.1 Introduction ............................................ 89 4.2 Existence of a Unique Solution ............................ 91 4.3 Properties of Solutions to Stochastic Differential Equations ... 93 4.4 Itˆo’s Formula and Exact Solutions ......................... 95 4.5 Approximating Stochastic Differential Equations ............ 99 4.6 Systems of Stochastic Differential Equations ................107 4.7 Forward Kolmogorov (Fokker-Planck) Equation .............109 4.8 Stability................................................111 4.9 Parameter Estimation for Stochastic Differential Equations ...118 4.9.1 A maximum likelihood estimation method ............118 4.9.2 A nonparametric estimation method .................121 Exercises ...................................................123 Computer Programs..........................................127 5 Modeling ..................................................135 5.1 Introduction ............................................135 5.2 Population Biology Examples .............................145 5.2.1 General model of two interacting populations .........145 5.2.2 Epidemic model and predator-prey model ............147 5.2.3 Persistence-time estimation .........................150 5.2.4 A population model with a time delay ...............152 5.2.5 A model including environmental variability ..........153 5.3 Physical Systems ........................................156 5.3.1 Mechanical vibration...............................156 5.3.2 Seed dispersal.....................................158 5.3.3 Ion transport .....................................160 5.3.4 Nuclear reactor kinetics ............................161 5.3.5 Precipitation......................................165 5.3.6 Chemical reactions ................................166 5.3.7 Cotton fiber breakage ..............................169 5.4 Some Stochastic Finance Models ..........................174 5.4.1 A stock-price model ...............................174 5.4.2 Option pricing ....................................177 5.4.3 Interest rates .....................................180 5.5 A Goodness-of-Fit Test for an SDE Model..................183 5.6 Alternate Equivalent SDE Models .........................186 Exercises ...................................................193 Computer Programs..........................................199 Contents ix References.....................................................217 Basic Notation ................................................223 Index..........................................................225 Preface The purpose of this book is to provide an introduction to the theory, compu- tation,andapplicationofItoˆstochasticdifferentialequations.Inparticular,a procedure for developing stochastic differential equation models is described and illustrated for applications in population biology, physics, and mathe- maticalfinance.Themodelingprocedureinvolvesfirstconstructingadiscrete stochasticprocessmodel.Thediscretemodelisdevelopedbystudyingchanges intheprocessoverasmalltimeinterval.AnItoˆstochasticdifferentialequation model is then formulated from the discrete stochastic model. The procedure isstraightforwardandisusefulformanydynamicalprocessesthatexperience random influences. The main topics in the theory and application of stochastic differential equations include random variables, stochastic processes, stochastic integra- tion,stochasticdifferentialequations,andmodels.Thesetopicsareintroduced andexaminedinseparatechapters.Manyexamplesaredescribedtoillustrate the concepts. The emphasis in the explanations is to provide a good under- standing of the concepts. Results are not necessarily presented in their most general form. Simplicity of presentation is chosen over generality. For the first four chapters, the theory of random processes and stochastic differen- tial equations is presented in a Hilbert space setting. A Hilbert space setting is chosen to unify and simplify the presentation of the material. The last chapter concentrates on explaining a model development procedure that is useful for constructing stochastic differential equation models for many kinds ofdynamicalsystemsexperiencingrandomchanges.Theprocedureproduces, inanaturalmanner,anItoˆstochasticdifferentialequationmodel,incontrast with, for example, a Stratonovich stochastic differential equation model. There are many excellent books available on the theory, application, and numericaltreatmentofstochasticdifferentialequations.Thebibliographylists manyofthesebooks.Itishopedthatthepresentbookwillcomplementthese previous books in providing an introduction to the development and testing of stochastic differential equation models. xi xii Preface Oneoftheobjectivesofthisbookistoprovideabasicintroductiontothe theory, approximation, and application of stochastic differential equations for anyone who is interested in pursuing research in this area. The intent of this book is to provide a background to stochastic differential equations so that the reader will be in a position to understand the mathematical literature in this area, including more advanced texts. To understand the material pre- sentedinthisbook,proficiencyinprobabilitytheoryanddifferentialequations is assumed. In particular, prerequisite courses for thoroughly understanding the concepts in this book include probability theory or statistics, differential equations,andintermediateanalysis.Inaddition,someknowledgeofscientific computing and programming would be very helpful. Throughout the book, approximation procedures are described. Problems involving stochastic inte- gration and stochastic differential equations can rarely be solved exactly and numerical procedures must be employed. In each chapter, one or two com- puter programs are listed in the computer languages MATLAB or Fortran. Each program is useful in solving a representative stochastic problem. The computer programs are listed in the book for convenience and to illustrate thattheprogramsaregenerallyshortandstraightforward.Attheendofeach chapter, analytical and computational exercises are provided. Several addi- tional computer programs are listed in these exercise sets. The exercises are designed to complement the material in the text. I am grateful to Texas Tech University for providing me a one-semester faculty development leave to write much of this book and the opportunity later to use this book in teaching a one-semester graduate course. I thank my wife, Linda Allen, for her encouragement on the writing of this book and for her many helpful comments and suggestions. I thank my friends and colleagues, Robert Paige, Henri Schurz, and Zhimin Zhang for their many positivesuggestionsandcriticismsonthemanuscript.IthankLynnBrandon, Springer Mathematics Publishing Editor, for her efficient assistance in the publication process. I am grateful to the several anonymous reviewers of the manuscriptfortheirpositivecommentsandtheirrecommendedrevisionsand additions. Finally, I am grateful to the colleagues and graduate students who worked with me on research projects. Lubbock, Texas, December 2006 Edward Allen