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1、AnIntroductiontoStochasticDifferentialEquationsVersion1.2LawrenceC.EvansDepartmentofMathematicsUCBerkeleyChapter1:IntroductionChapter2:AcrashcourseinbasicprobabilitytheoryChapter3:Brownianmotionand“whitenoise”Chapter4:Stochasticintegrals,Itˆo’sformulaChapter5:S
2、tochasticdifferentialequationsChapter6:ApplicationsAppendicesExercisesReferences1PREFACEThesenotessurvey,withouttoomanyprecisedetails,thebasictheoryofprob-ability,randomdifferentialequationsandsomeapplications.Stochasticdifferentialequationsisusually,andjustly,re
3、gardedasagraduatelevelsubject.Areallycarefultreatmentassumesthestudents’familiaritywithprobabilitytheory,measuretheory,ordinarydifferentialequations,andpartialdif-ferentialequationsaswell.ButasanexperimentItriedtodesigntheselecturessothatstartinggraduatestudent
4、s(andmaybereallystrongundergraduates)canfollowmostofthetheory,atthecostofsomeomissionofdetailandprecision.Iforinstancedownplayedmostmeasuretheoreticissues,butdidemphasizetheintuitiveideaofσ–algebrasas“containinginformation”.Similarly,I“prove”manyformulasbyconfi
5、rmingthemineasycases(forsimplerandomvariablesorforstepfunctions),andthenjuststatingthatbyapproximationtheserulesholdingeneral.Ialsodidnotreproduceinclasssomeofthemorecomplicatedproofsprovidedinthesenotes,althoughIdidtrytoexplaintheguidingideas.Mythanksespecial
6、lytoLisaGoldberg,whoseveralyearsagopresentedmyclasswithseverallecturesonfinancialapplications,andtoFraydounRezakhanlou,whohastaughtfromthesenotesandaddedseveralimprovements.IamalsogratefultoJonathanWeareforseveralcomputersimulationsillus-tratingthetext.Thanksal
7、sotomanyreaderswhohavefounderrors,especiallyRobertPiche,whoprovidedmewithanextensivelistoftyposandsuggestionsthatIhaveincorporatedintothislatestversionofthenotes.2CHAPTER1:INTRODUCTIONA.MOTIVATIONFixapointx∈Rnandconsiderthentheordinarydifferentialequation:0x˙(
8、t)=b(x(t))(t>0)(ODE)x(0)=x0,whereb:Rn→Rnisagiven,smoothvectorfieldandthesolutionisthetrajectoryx(·):[0,∞)→Rn.x(t)x0TrajectoryofthedifferentialequationNotation.x(t)isthestateofthesys
