Diffusion Processes, Jump Processes, and Stochastic Differential Equations

139 Pages·2022·8.183 MB·English

by  Wojbor A. Woyczyński

Diffusion Processes, Jump
Processes, and Stochastic
Differential Equations

Diffusion Processes, Jump
Processes, and Stochastic
Differential Equations
Wojbor A. Woyczyn´ski
Case Western Reserve University, USA
Firsteditionpublished2022
byCRCPress
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Library of Congress Cataloging-in-Publication Data
Names:Woyczyn´ski,W.A.(WojborAndrzej),1943-author.
Title:Diffusionprocesses,jumpprocesses,andstochasticdifferential
equations/WojborA.Woyczyn´ski,CaseWesternReserveUniversity,USA.
Description:Firstedition.|BocaRaton:Chapman&Hall/CRCPress,2022.|
Includesbibliographicalreferencesandindex.
Identifiers:LCCN2021032255(print)|LCCN2021032256(ebook)|
ISBN9781032100678(hardback)|ISBN9781032107271(paperback)|
ISBN9781003216759(ebook)
Subjects:LCSH:Diffusionprocesses.|Jumpprocesses.|
Stochasticdifferentialequations.
Classification:LCCQA274.75.W692022(print)|LCCQA274.75(ebook)|
DDC519.2/33—dc23
LCrecordavailableathttps://lccn.loc.gov/2021032255
LCebookrecordavailableathttps://lccn.loc.gov/2021032256
ISBN:9781032100678(hbk)
ISBN:9781032107271(pbk)
ISBN:9781003216759(ebk)
DOI:10.1201/9781003216759
TypesetinMinion
bycodeMantra
The children and I so want to dedicate the book to:
The Students of Prof. Wojbor A. Woyczyn´ski
Prof. Paromita Banerjee, his last PhD graduate, and her husband, Prof. Anirban
Mondal, his colleague at Case Western Reserve University, assisted with the posthu-
mous preparations required for publication.
—Elizabeth H. Woyczyn´ski

Contents
AUTHOR xi
Chapter
1■ Random Variables, Vectors, Processes, and Fields 1
1.1 RANDOM VARIABLES, VECTORS, AND THEIR DISTRIBUTIONS—A
GLOSSARY 1
1.1.1 Basic Concepts 1
1.1.2 Absolutely Continuous, Discrete, Mixed, and Singular
Probability Distributions 3
1.1.3 CharacteristicFunctions,LaplaceTransforms,and
Moment-Generating Functions 3
1.1.4 Examples 4
1.2 LAWOFLARGENUMBERSANDTHECENTRALLIMITTHEOREM 7
1.3 STOCHASTICPROCESSESANDTHEIRFINITE-DIMENSIONAL
DISTRIBUTIONS 9
1.4 PROBLEMSANDEXERCISES 10
Chapter
2■ From Random Walk to Brownian Motion 11
2.1 SYMMETRICRANDOMWALK;PARABOLICRESCALINGAND
RELATEDFOKKER-PLANCKEQUATIONS 11
2.1.1 Brownian Motion as Hydrodynamic Limit of Random Walks 11
2.1.2 Brownian Motion via the Central Limit Theorem and the
Invariance Principle 14
2.2 BASICPROPERTIESOFBROWNIANMOTION 16
2.3 ALMOSTSURECONTINUITYOFSAMPLEPATHS 18
2.4 NOWHEREDIFFERENTIABILITYOFBROWNIANMOTION 21
2.5 HITTINGTIMES,ANDOTHERSUBTLEPROPERTIESOFBROWNIAN
MOTION 23
2.6 PROBLEMSANDEXERCISES 24
vii
viii ■ Contents
Chapter
3■ Poisson Processes and Their Mixtures 27
3.1 WHYPOISSONPROCESS? 27
3.2 COVARIANCE STRUCTURE AND FINITE DIMENSIONAL
DISTRIBUTIONS 29
3.3 WAITINGTIMESANDINTER-JUMPTIMES 30
3.4 EXTENSIONSANDGENERALIZATIONS 34
3.5 FRACTIONALPOISSONPROCESSES(fPp) 35
3.5.1 FPp Interarrival Time 35
3.6 PROBLEMSANDEXERCISES 42
Chapter 4■ L´evy Processes and the L´evy-Khinchine Formula: Basic Facts 43
4.1 PROCESSESWITHSTATIONARYANDINDEPENDENTINCREMENTS 43
´
4.2 FROMPOISSONPROCESSESTOLEVYPROCESSES 44
´
4.3 INFINITESIMALGENERATORSOFLEVYPROCESSES 46
´
4.4 SELF-SIMILARLEVYPROCESSES 47
4.5 PROPERTIESOFα-STABLEMOTIONS 48
4.6 INFINITESIMALGENERATORSOFα-STABLEMOTIONS 50
4.7 PROBLEMSANDEXERCISES 51
Chapter
5■ General Processes with Independent Increments 53
5.1 NONSTATIONARYPROCESSESWITHINDEPENDENTINCREMENTS 53
5.2 STOCHASTICCONTINUITYANDJUMPPROCESSES 55
5.3 ANALYSISOFJUMPSTRUCTURE 57
5.4 RANDOM MEASURES AND RANDOM INTEGRALS ASSOCIATED
WITHJUMPPROCESSES 58
5.4.1 Random Measures and Random Integrals 58
5.5 STRUCTUREOFGENERALI.I.PROCESSES 59
Chapter
6■ StochasticIntegralsforBrownianMotionandGeneral
´
Levy Processes 65
6.1 WIENERRANDOMINTEGRAL 65
6.2 ITOˆ’SSTOCHASTICINTEGRALFORBROWNIANMOTION 66
6.3 ANINSTRUCTIVEEXAMPLE 70
6.4 ITOˆ’SFORMULA 71
Contents ■ ix
6.5 MARTINGALEPROPERTYOFITOˆ INTEGRALS 73
6.6 WIENER AND ITOˆ-TYPE STOCHASTIC INTEGRALS FOR α-STABLE
MOTIONANDGENERALLE´VYPROCESSES 74
Chapter 7■ Itˆo Stochastic Differential Equations 77
7.1 DIFFERENTIALEQUATIONSWITHNOISE 77
7.2 STOCHASTICDIFFERENTIALEQUATIONS:BASICTHEORY 79
7.3 SDEsWITHCOEFFICIENTSDEPENDINGONLYONTIME 84
7.4 POPULATIONGROWTHMODELANDOTHEREXAMPLES 85
7.4.1 Population Growth Model 85
7.4.2 Ornstein–Uhlenbeck Process 86
7.5 SYSTEMSOFSDEsANDVECTOR-VALUEDITOˆ’SFORMULA 86
7.6 NUMERICAL SOLUTION OF STOCHASTIC DIFFERENTIAL
EQUATIONS 89
Chapter
8■ Asymmetric Exclusion Processes and Their Scaling Limits 91
8.1 ASYMMETRICEXCLUSIONPRINCIPLES 91
8.2 SCALINGLIMIT 92
8.3 OTHER QUEUING REGIMES RELATED TO NON-NEAREST
NEIGHBORSYSTEMS 93
8.4 NETWORKS WITH MULTISERVER NODES AND PARTICLE SYSTEMS
WITHSTATE-DEPENDENTRATES 97
8.5 SHOCKANDRAREFACTIONWAVESOLUTIONSFORTHERIEMANN
PROBLEMFORCONSERVATIONLAWS 102
Chapter
9■ Nonlinear Diffusion Equations 107
9.1 HYPERBOLICEQUATIONS 107
9.2 NONLINEARDIFFUSIONAPPROXIMATIONS 109
9.3 NONLINEARPROCESSES 113
9.4 INTERACTINGDIFFUSIONSANDMONTE-CARLOMETHODS 114
Appendix
A■ The Remarkable Bernoulli Family 115
Bibliography 119
Index 123