introduction to stochastic differential equations v1.2 (berkeley lecture notes) - l. evans

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

2、erentialequationsChapter6:ApplicationsExercisesAppendicesReferences1PREFACETheseareanevolvingsetofnotesforMathematics195atUCBerkeley.Thiscourseisforadvancedundergraduatemathmajorsandsurveyswithouttoomanyprecisedetailsrandomdifferentialequationsandsomeapplications.Stochastic

3、differentialequationsisusually,andjustly,regardedasagraduatelevelsubject.Areallycarefultreatmentassumesthestudents’familiaritywithprobabilitytheory,measuretheory,ordinarydifferentialequations,andperhapspartialdifferentialequationsaswell.Thisisalltoomuchtoexpectofundergrads.Bu

4、twhitenoise,Brownianmotionandtherandomcalculusarewonderfultopics,toogoodforundergraduatestomissouton.ThereforeasanexperimentItriedtodesigntheselecturessothatstrongstudentscouldfollowmostofthetheory,atthecostofsomeomissionofdetailandprecision.Iforinstancedownplayedmostmeasu

5、retheoreticissues,butdidemphasizetheintuitiveideaofσ–algebrasas“containinginformation”.Similarly,I“prove”manyformulasbyconfirmingthemineasycases(forsimplerandomvariablesorforstepfunctions),andthenjuststatingthatbyapproximationtheserulesholdingeneral.Ialsodidnotreproduceincl

6、asssomeofthemorecomplicatedproofsprovidedinthesenotes,althoughIdidtrytoexplaintheguidingideas.MythanksespeciallytoLisaGoldberg,whoseveralyearsagopresentedtheclasswithseverallecturesonfinancialapplications,andtoFraydounRezakhanlou,whohastaughtfromthesenotesandaddedseveralimp

7、rovements.IamalsogratefultoJonathanWeareforseveralcomputersimulationsillustratingthetext.2CHAPTER1:INTRODUCTIONA.MOTIVATIONFixapointx∈Rnandconsiderthentheordinarydifferentialequation:0x˙(t)=b(x(t))(t>0)(ODE)x(0)=x0,whereb:Rn→Rnisagiven,smoothvectorfieldandthesolutionisthetr

8、ajectoryx(·):[0,∞)→Rn.

TrajectoryofthedifferentialequationNotation.x(t)isthestateoft