随机过程(斯坦福大学)

StochasticProcessesAmirDembo(revisedbyKevinRoss)April8,2008E-mailaddress:[e-mail]s,StanfordUniversity,Stanford,CA94305.ContentsPreface5Chapter1.Probability,measureandintegration71.1.Probabilityspacesand-elds71.2.Randomvariablesandtheirexpectation101.3.Convergenceofrandomvariables191.4.Independence,weakconvergenceanduniformintegrability25Chapter2.ConditionalexpectationandHilbertspaces352.1.Conditionalexpectation:existenceanduniqueness352.2.Hilbertspaces392.3.Propertiesoftheconditionalexpectation432.4.Regularconditionalprobability46Chapter3.StochasticProcesses:generaltheory493.1.Denition,distributionandversions493.2.Characteristicfunctions,Gaussianvariablesandprocesses553.3.Samplepathcontinuity62Chapter4.Martingalesandstoppingtimes674.1.Discretetimemartingalesandltrations674.2.Continuoustimemartingalesandrightcontinuousltrations734.3.Stoppingtimesandtheoptionalstoppingtheorem764.4.Martingalerepresentationsandinequalities824.5.Martingaleconvergencetheorems884.6.Branchingprocesses:extinctionprobabilities90Chapter5.TheBrownianmotion955.1.Brownianmotion:denitionandconstruction955.2.ThereectionprincipleandBrownianhittingtimes1015.3.SmoothnessandvariationoftheBrowniansamplepath103Chapter6.Markov,PoissonandJumpprocesses1116.1.Markovchainsandprocesses1116.2.Poissonprocess,Exponentialinter-arrivalsandorderstatistics1196.3.Markovjumpprocesses,compoundPoissonprocesses125Bibliography127Index1293PrefaceThesearethelecturenotesforaonequartergraduatecourseinStochasticPro-cessesthatItaughtatStanfordUniversityin2002and2003.ThiscourseisintendedforincomingmasterstudentsinStanford'sFinancialMathematicsprogram,forad-vancedundergraduatesmajoringinmathematicsandforgraduatestudentsfromEngineering,Economics,StatisticsortheBusinessschool.OnepurposeofthistextistopreparestudentstoarigorousstudyofStochasticDierentialEquations.Morebroadly,itsgoalistohelpthereaderunderstandthebasicconceptsofmeasurethe-orythatarerelevanttothemathematicaltheoryofprobabilityandhowtheyapplytotherigorousconstructionofthemostfundamentalclassesofstochasticprocesses.Towardsthisgoal,weintroduceinChapter1therelevantelementsfrommeasureandintegrationtheory,namely,theprobabilityspaceandthe-eldsofeventsinit,randomvariablesviewedasmeasurablefunctions,theirexpectationasthecorrespondingLebesgueintegral,independence,distributionandvariousnotionsofconvergence.ThisissupplementedinChapter2bythestudyoftheconditionalexpectation,viewedasarandomvariabledenedviathetheoryoforthogonalprojectionsinHilbertspaces.AfterthisexplorationofthefoundationsofProbabilityTheory,weturninChapter3tothegeneraltheoryofStochasticProcesses,withaneyetowardsprocessesindexedbycontinuoustimeparametersuchastheBrownianmotionofChapter5andtheMarkovjumpprocessesofChapter6.Havingthisinmind,Chapter3isaboutthenitedimensionaldistributionsandtheirrelationtosamplepathcontinuity.AlongthewaywealsointroducetheconceptsofstationaryandGaussianstochasticprocesses.Chapter4dealswithltrations,themathematicalnotionofinformationpro-gressionintime,andwiththeassociatedcollectionofstochasticprocessescalledmartingales.Wetreatbothdiscreteandcontinuoustimesettings,emphasizingtheimportanceofright-continuityofthesamplepathandltrationinthelattercase.Martingalerepresentationsareexplored,aswellasmaximalinequalities,conver-gencetheoremsandapplicationstothestudyofstoppingtimesandtoextinctionofbranchingprocesses.Chapter5providesanintroductiontothebeautifultheoryoftheBrownianmo-tion.ItisrigorouslyconstructedhereviaHilbertspacetheoryandshowntobeaGaussianmartingaleprocessofstationaryindependentincrements,withcontinuoussamplepathandpossessingthestrongMarkovproperty.Fewofthemanyexplicitcomputationsknownforthisprocessarealsodemonstrated,mostlyinthecontextofhittingtimes,runningmaximaandsamplepathsmoothnessandregularity.56PREFACEChapter6providesabriefintroductiontothetheoryofMarkovchainsandpro-cesses,avastsubjectatthecoreofprobabilitytheory,towhichmanytextbooksaredevoted.WeillustratesomeoftheinterestingmathematicalpropertiesofsuchprocessesbyexaminingthespecialcaseofthePoissonprocess,andmoregenerally,thatofMarkovjumpprocesses.Asclearfromthepreceding,itnormallytakesmorethanayeartocoverthescopeofthistext.Evenmoreso,giventhattheintendedaudienceforthiscoursehasonlyminimalpriorexposuretostochasticprocesses(beyondtheusualelementaryprob-abilityclasscoveringonlydiscretesettingsandvariableswithprobabilitydensityfunction).WhilestudentsareassumedtohavetakenarealanalysisclassdealingwithRiemannintegration,nopriorknowledgeofmeasuretheoryisassumedhere.Theunusualsolutiontothissetofconstraintsistoproviderigorousdenitions,examplesandtheoremstatements,whileforgoingtheproofsofallbutthemosteasyderivations.Atthissomewhatsuperciallevel,onecancovereverythinginaonesemestercourseoffortylecturehours(andifonehashighlymotivatedstudentssuchasIhadinStanford,evenaonequartercourseofthirtylecturehoursmightwork).InpreparingthistextIwasmuchinuencedbyZakai'sunpublishedlecturenotes[Zak].RevisedandexpandedbyShwartzandZeitouniitisusedtothisday