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StochasticProcessesAmirDembo(revisedbyKevinRoss)April8,2008E-mailaddress:amir@stat.stanford.eduDepartmentofStatistics,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.OnepurposeofthistextistopreparestudentstoarigorousstudyofStochasticDi erentialEquations.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