Modeling with Ito Stochastic Differential Equations

237 Pages·2007·1.026 MB·English

by  Allen E.

Modeling with Itoˆ Stochastic Differential Equations
MATHEMATICAL MODELLING:
Theory and Applications
VOLUME 22
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Modeling with Itoˆ
Stochastic Differential
Equations
By
E. Allen
Texas Tech University, USA
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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