An introduction to stochastic differential equations.pdf

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1、AnIntroductiontoStochasticDi?erentialEquationsVersion1.2LawrenceC.EvansDepartmentofMathematicsUCBerkeleyChapter1:IntroductionChapter2:AcrashcourseinbasicprobabilitytheoryChapter3:Brownianmotionand“whitenoise”Chapter4:Stochasticintegrals,It?o’sformulaChapter5:S

2、tochasticdi?erentialequationsChapter6:ApplicationsAppendicesExercisesReferences1PREFACEThesenotessurvey,withouttoomanyprecisedetails,thebasictheoryofprob-ability,randomdi?erentialequationsandsomeapplications.Stochasticdi?erentialequationsisusually,andjustly,re

3、gardedasagraduatelevelsubject.Areallycarefultreatmentassumesthestudents’familiaritywithprobabilitytheory,measuretheory,ordinarydi?erentialequations,andpartialdif-ferentialequationsaswell.ButasanexperimentItriedtodesigntheselecturessothatstartinggraduatestudent

4、s(andmaybereallystrongundergraduates)canfollowmostofthetheory,atthecostofsomeomissionofdetailandprecision.Iforinstancedownplayedmostmeasuretheoreticissues,butdidemphasizetheintuitiveideaofσ–algebrasas“containinginformation”.Similarly,I“prove”manyformulasbycon?

5、rmingthemineasycases(forsimplerandomvariablesorforstepfunctions),andthenjuststatingthatbyapproximationtheserulesholdingeneral.Ialsodidnotreproduceinclasssomeofthemorecomplicatedproofsprovidedinthesenotes,althoughIdidtrytoexplaintheguidingideas.Mythanksespecial

6、lytoLisaGoldberg,whoseveralyearsagopresentedmyclasswithseverallectureson?nancialapplications,andtoFraydounRezakhanlou,whohastaughtfromthesenotesandaddedseveralimprovements.IamalsogratefultoJonathanWeareforseveralcomputersimulationsillus-tratingthetext.Thanksal

7、sotomanyreaderswhohavefounderrors,especiallyRobertPiche,whoprovidedmewithanextensivelistoftyposandsuggestionsthatIhaveincorporatedintothislatestversionofthenotes.2CHAPTER1:INTRODUCTIONA.MOTIVATIONFixapointx∈Rnandconsiderthentheordinarydi?erentialequation:0x˙(

8、t)=b(x(t))(t>0)(ODE)x(0)=x0,whereb:Rn→Rnisagiven,smoothvector?eldandthesolutionisthetrajectoryx(·):[0,∞)→Rn.x(t)x0TrajectoryofthedifferentialequationNotation.x(t)isthestateofthesys