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1、ANINTRODUCTIONTOSTOCHASTICDIFFERENTIALEQUATIONSVERSION1.2LawrenceC.EvansDepartmentofMathematicsUCBerkeleyChapter1:IntroductionChapter2:AcrashcourseinbasicprobabilitytheoryChapter3:Brownianmotionand“whitenoise”Chapter4:Stochasticintegrals,It?o’sformulaChapter5:Stochasticdi?
2、erentialequationsChapter6:ApplicationsExercisesAppendicesReferences1PREFACETheseareanevolvingsetofnotesforMathematics195atUCBerkeley.Thiscourseisforadvancedundergraduatemathmajorsandsurveyswithouttoomanyprecisedetailsrandomdi?erentialequationsandsomeapplications.Stochastic
3、di?erentialequationsisusually,andjustly,regardedasagraduatelevelsubject.Areallycarefultreatmentassumesthestudents’familiaritywithprobabilitytheory,measuretheory,ordinarydi?erentialequations,andperhapspartialdi?erentialequationsaswell.Thisisalltoomuchtoexpectofundergrads.Bu
4、twhitenoise,Brownianmotionandtherandomcalculusarewonderfultopics,toogoodforundergraduatestomissouton.ThereforeasanexperimentItriedtodesigntheselecturessothatstrongstudentscouldfollowmostofthetheory,atthecostofsomeomissionofdetailandprecision.Iforinstancedownplayedmostmeasu
5、retheoreticissues,butdidemphasizetheintuitiveideaofσ–algebrasas“containinginformation”.Similarly,I“prove”manyformulasbycon?rmingthemineasycases(forsimplerandomvariablesorforstepfunctions),andthenjuststatingthatbyapproximationtheserulesholdingeneral.Ialsodidnotreproduceincl
6、asssomeofthemorecomplicatedproofsprovidedinthesenotes,althoughIdidtrytoexplaintheguidingideas.MythanksespeciallytoLisaGoldberg,whoseveralyearsagopresentedtheclasswithseverallectureson?nancialapplications,andtoFraydounRezakhanlou,whohastaughtfromthesenotesandaddedseveralimp
7、rovements.IamalsogratefultoJonathanWeareforseveralcomputersimulationsillustratingthetext.2CHAPTER1:INTRODUCTIONA.MOTIVATIONFixapointx∈Rnandconsiderthentheordinarydi?erentialequation:0x˙(t)=b(x(t))(t>0)(ODE)x(0)=x0,whereb:Rn→Rnisagiven,smoothvector?eldandthesolutionisthetr
8、ajectoryx(·):[0,∞)→Rn.TrajectoryofthedifferentialequationNotation.x(t)isthestateoft
