an introduction to stochastic differential equations (evans)

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1、ANINTRODUCTIONTOSTOCHASTICDIFFERENTIALEQUATIONSVERSION1.2LawrenceC.EvansDepartmentofMathematicsUCBerkeleyChapter1:IntroductionChapter2:AcrashcourseinbasicprobabilitytheoryChapter3:Brownianmotionand“whitenoise”Chapter4:Stochasticintegrals,Itˆo’sformulaChapter5:StochasticdifferentialequationsChapter6:

2、ApplicationsAppendicesExercisesReferences1PREFACEThesenotessurvey,withouttoomanyprecisedetails,thebasictheoryofprobability,randomdifferentialequationsandsomeapplications.Stochasticdifferentialequationsisusually,andjustly,regardedasagraduatelevelsubject.Areallycarefultreatmentassumesthestudents’famili

3、aritywithprobabilitytheory,measuretheory,ordinarydifferentialequations,andpartialdifferentialequationsaswell.ButasanexperimentItriedtodesigntheselecturessothatstartinggraduatestudents(andmaybereallystrongundergraduates)canfollowmostofthetheory,atthecostofsomeomissionofdetailandprecision.Iforinstanced

4、ownplayedmostmeasuretheoreticissues,butdidemphasizetheintuitiveideaofσ–algebrasas“containinginformation”.Similarly,I“prove”manyformulasbyconfirmingthemineasycases(forsimplerandomvariablesorforstepfunctions),andthenjuststatingthatbyapproximationtheserulesholdingeneral.Ialsodidnotreproduceinclasssomeo

5、fthemorecomplicatedproofsprovidedinthesenotes,althoughIdidtrytoexplaintheguidingideas.MythanksespeciallytoLisaGoldberg,whoseveralyearsagopresentedmyclasswithseverallecturesonfinancialapplications,andtoFraydounRezakhanlou,whohastaughtfromthesenotesandaddedseveralimprovements.IamalsogratefultoJonathan

6、Weareforseveralcomputersimulationsillustratingthetext.Thanksalsotomanyreaderswhohavefounderrors,especiallyRobertPiche,whoprovidedmewithanextensivelistoftyposandsuggestionsthatIhaveincorporatedintothislatestversionofthenotes.2CHAPTER1:INTRODUCTIONA.MOTIVATIONFixapointx∈Rnandconsiderthentheordinarydi

7、fferentialequation:0x˙(t)=b(x(t))(t>0)(ODE)x(0)=x0,whereb:Rn→Rnisagiven,smoothvectorfieldandthesolutionisthetrajectoryx(·):[0,∞)→Rn.

TrajectoryofthedifferentialequationNotation.x(t)isthestateofthesystem