Stochastic Analysis and Diffusion Processes

365 Pages·2014·1.884 MB·English

by  Gopinath Kallianpur, P Sundar

OXFORD GRADUATE TEXTS IN MATHEMATICS
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Stochastic Analysis
and Diffusion Processes
gopinath kallianpur and p. sundar
3
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Preface
Theideaofwritingabookonstochasticanalysisarosefromasuggestionthatwewrite
anenjoyablebookonBrownianmotionandstochasticanalysis.Afterstartingour
work,wefeltthattheword“enjoyable”shouldnotbeconstruedasasynonymfor“heur-
isticandnon-rigorous”;rather,itshouldmean“readableandconcisewithfulldetails”.
Writingsuchabookisatallorder,andatbest,onlypartialsuccessisallthatonecan
hopefor.
Theoriginofthebookowesitselftoourlecturenotesformedoveranumberofyears
anddrawnfrommanyinspiringsources.Inthepresentbook,webuildthebasictheory
ofstochasticcalculusinaself-containedmannerinthefirstsixchapters.Startingwith
Kolmogorov’sconstructionofstochasticprocesses,Brownianmotionandmartingales
are presented with a view to build stochastic integration theory and study stochastic
differentialequations.Thiswouldconstitutethefirstpartofthebook.
Thenextsixchaptersdealwiththeprobabilisticbehaviorofdiffusionprocessesand
certainfineraspects,applications,andextensionsofthetheory.Onecanviewitasthe
second part of this book. The selection of material for the second part of the book
reflectsourowntastesandprovidesonlyaglimpseofsomeoftheactiveareasofresearch.
Stochasticanalysisbeingsovast,importanttopicssuchasMalliavincalculus,stochastic
control,andfilteringtheory,thoughofinteresttous,hadtobeleftout.Instead,westart
withmartingaleproblems,amethoduniquetostochasticanalysis,andproceedtodis-
cuss the connection between stochastic analysis and partial differential equations and
thenstudyGaussiansolutionsofstochasticequations.JumpMarkovprocesses,invari-
ant measures, and large deviations principle for diffusions are presented in successive
chapters,thougheachofthesecaneasilyformthesubjectmatterforawholebook.
Thebookiswrittenforgraduatestudents,appliedmathematicians,andanyoneinter-
ested in learning stochastic calculus. The reader is assumed to be knowledgeable in
probabilitytheoryatagraduatelevel.Acourseonstochasticanalysiscanbedesigned
usingthefirstpartofthisbookalongwithpartsofChapter8.Aselectionofthelastsix
chapterscanbeusedforasecondcourseonstochasticanalysis.
Regardinginterdependenceofchapters,thefirstpartofthebookisconnectednatur-
allyasasequencewithoneexception:Chapter4isnotneededtoreadChapters5and
6.Agoodknowledgeofthefirstpartisrequiredtoreadanyofthefollowingchapters.
However,thesecondpartaffordsmoreflexibilityinreading.Forinstance,eachofthe
Chapters9,10,and12canbereadindependentlyoftheothers.
viii | Preface
Wearequiteindebtedtoseveralofourfriendsandcolleagueswhohelpedandinspired
ustowritethisbook.Wethankseveralofourcolleagues,especiallyG.Ferreyra,H.-H.
Kuo,U.Manna,P.E.Protter,B.Rüdiger,A.Sengupta,S.S.Sritharan,W.Woyczynski,
andH.Yin.Severalofourgraduatestudentsreadpartsofthebookandspottednumerous
typos.Weappreciatetheireffortsandthankthemfortheircarefulreading.Wethankour
families,especiallyKrishna,Kathy,andVijayforbeingpatientwithusandhelpingus
cheerfullywhilethebookwaswritten.ThankstoMs.ElizabethFarrellforathorough
proofreadingoftheentiremanuscript.
Contents
1 IntroductiontoStochasticProcesses 1
1.1 TheKolmogorovConsistencyTheorem 1
1.2 TheLanguageofStochasticProcesses 11
1.3 SigmaFields,Measurability,andStoppingTimes 14
Exercises 17
2 BrownianMotion 19
2.1 DefinitionandConstructionofBrownianMotion 20
2.2 EssentialFeaturesofaBrownianMotion 27
2.3 TheReflectionPrinciple 34
Exercises 39
3 ElementsofMartingaleTheory 41
3.1 DefinitionandExamplesofMartingales 41
3.2 WienerMartingalesandtheMarkovProperty 44
3.3 EssentialResultsonMartingales 49
3.4 TheDoob-MeyerDecomposition 54
3.5 TheMeyerProcessforL2-martingales 67
3.6 LocalMartingales 71
Exercises 73
4 AnalyticalToolsforBrownianMotion 75
4.1 Introduction 75
4.2 TheBrownianSemigroup 76
4.3 ResolventsandGenerators 79
4.4 PregeneratorsandMartingales 87
Exercises 89
5 StochasticIntegration 90
5.1 TheItôIntegral 90
5.2 PropertiesoftheIntegral 98
5.3 Vector-valuedProcesses 105
5.4 TheItôFormula 106