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Diffusion Processes, Jump Processes, and Stochastic Differential Equations PDF

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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 6000BrokenSoundParkwayNW,Suite300,BocaRaton,FL33487-2742 andbyCRCPress 2ParkSquare,MiltonPark,Abingdon,Oxon,OX144RN ©2022WojborA.Woyczyn´ski CRCPressisanimprintofTaylor&FrancisGroup,LLC Reasonableeffortshavebeenmadetopublishreliabledataandinformation,buttheauthorandpublisher cannotassumeresponsibilityforthevalidityofallmaterialsortheconsequencesoftheiruse.Theauthors andpublishershaveattemptedtotracethecopyrightholdersofallmaterialreproducedinthispublication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyrightmaterialhasnotbeenacknowledgedpleasewriteandletusknowsowemayrectifyinanyfuture reprint. ExceptaspermittedunderU.S.CopyrightLaw,nopartofthisbookmaybereprinted,reproduced,trans- mitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system,withoutwrittenpermissionfromthepublishers. For permission to photocopy or use material electronically from this work, access www.copyright.com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750- 8400.ForworksthatarenotavailableonCCCpleasecontactmpkbookspermissions@tandf.co.uk Trademarknotice:Productorcorporatenamesmaybetrademarksorregisteredtrademarks,andareused onlyforidentificationandexplanationwithoutintenttoinfringe. 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

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