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1、UniversitextEditorialBoard(NorthAmerica):S.AxlerK.A.RibetHui-HsiungKuoIntroductiontoStochasticIntegrationHui-HsiungKuoDepartmentofMathematicsLouisianaStateUniversityBatonRouge,LA70803-4918USAkuo@math.lsu.eduEditorialBoard(NorthAmerica):S.AxlerK.A.RibetMathem
2、aticsDepartmentMathematicsDepartmentSanFranciscoStateUniversityUniversityofCaliforniaatBerkeleySanFrancisco,CA94132Berkeley,CA94720-3840USAUSAaxler@sfsu.eduribet@math.berkeley.eduMathematicsSubjectClassification(2000):60-XXLibraryofCongressControlNumber:2005
3、935287ISBN-10:0-387-28720-5Printedonacid-freepaper.ISBN-13:978-0387-28720-1©2006SpringerScience+BusinessMedia,Inc.Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewrittenpermissionofthepublisher(SpringerScience+BusinessMedia,Inc.,
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5、pedisforbidden.Theuseinthispublicationoftradenames,trademarks,servicemarks,andsimilarterms,eveniftheyarenotidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyaresubjecttoproprietaryrights.PrintedintheUnitedStatesofAmerica.(EB)987654321
6、springeronline.comDedicatedtoKiyosiItˆo,andinmemoryofhiswifeShizueItˆoPrefaceIntheLeibniz–Newtoncalculus,onelearnsthedifferentiationandintegrationofdeterministicfunctions.Abasictheoremindifferentiationisthechainrule,whichgivesthederivativeofacompositeoftwodiffe
7、rentiablefunctions.Thechainrule,whenwritteninanindefiniteintegralform,yieldsthemethodofsubstitution.Inadvancedcalculus,theRiemann–Stieltjesintegralisdefinedthroughthesameprocedureof“partition-evaluation-summation-limit”asintheRiemannintegral.Indealingwithrando
8、mfunctionssuchasfunctionsofaBrownianmotion,thechainrulefortheLeibniz–Newtoncalculusbreaksdown.ABrownianmotionmovessorapidlyandirregularlythatalmostallofitssamplepathsarenowheredifferentiable.Thus
