Stochastic Processes

Download Stochastic Processes

Post on 22-Oct-2014

110 views

Category:

Documents

0 download

Embed Size (px)

TRANSCRIPT

<p>StochasticProcessesAmirDembo(revisedbyKevinRoss)April8,2008E-mail address: amir@stat.stanford.eduDepartmentofStatistics, StanfordUniversity,Stanford,CA94305.ContentsPreface 5Chapter1. Probability,measureandintegration 71.1. Probabilityspacesand-elds 71.2. Randomvariablesandtheirexpectation 101.3. Convergenceofrandomvariables 191.4. Independence,weakconvergence anduniformintegrability 25Chapter2. ConditionalexpectationandHilbertspaces 352.1. Conditionalexpectation: existenceanduniqueness 352.2. Hilbertspaces 392.3. Propertiesoftheconditionalexpectation 432.4. Regularconditionalprobability 46Chapter3. StochasticProcesses: generaltheory 493.1. Denition,distributionandversions 493.2. Characteristicfunctions,Gaussianvariablesandprocesses 553.3. Samplepathcontinuity 62Chapter4. Martingalesandstoppingtimes 674.1. Discretetimemartingalesandltrations 674.2. Continuoustimemartingalesandrightcontinuousltrations 734.3. Stoppingtimesandtheoptionalstoppingtheorem 764.4. Martingalerepresentations andinequalities 824.5. Martingaleconvergence theorems 884.6. Branchingprocesses: extinctionprobabilities 90Chapter5. TheBrownian motion 955.1. Brownian motion: denitionandconstruction 955.2. ThereectionprincipleandBrownian hittingtimes 1015.3. SmoothnessandvariationoftheBrownian samplepath 103Chapter6. Markov,Poisson andJumpprocesses 1116.1. Markovchainsandprocesses 1116.2. Poisson process,Exponentialinter-arrivals andorderstatistics 1196.3. Markovjumpprocesses,compoundPoisson processes 125Bibliography 127Index 1293PrefaceThesearethelecturenotesforaonequartergraduatecourseinStochasticPro-cesses that I taught at Stanford University in 2002 and 2003. This course is intendedfor incoming master students in Stanfords Financial Mathematics program, for ad-vancedundergraduatesmajoringinmathematicsandforgraduatestudentsfromEngineering, Economics, Statistics or the Business school. One purpose of this textis to prepare students to a rigorous study of Stochastic Dierential Equations. Morebroadly, its goal is to help the reader understand the basic concepts of measure the-ory that are relevant to the mathematical theory of probability and how they applyto the rigorous construction of the most fundamental classes of stochastic processes.Towards thisgoal,weintroduceinChapter1therelevant elementsfrommeasureandintegrationtheory, namely, theprobabilityspaceandthe-elds of eventsinit, randomvariablesviewedasmeasurablefunctions, theirexpectationasthecorresponding Lebesgue integral, independence,distribution and various notions ofconvergence. ThisissupplementedinChapter2bythestudyof theconditionalexpectation, viewedas arandomvariable denedviathe theoryof orthogonalprojectionsinHilbertspaces.After this exploration of the foundations of Probability Theory, we turn in Chapter3to the general theoryof Stochastic Processes, withaneye towards processesindexedbycontinuoustimeparametersuchastheBrownianmotionof Chapter5andtheMarkovjumpprocesses of Chapter 6. Havingthis inmind, Chapter3is aboutthenitedimensional distributionsandtheirrelationtosamplepathcontinuity. Along the way we also introduce the concepts of stationary and Gaussianstochasticprocesses.Chapter 4 deals withltrations, the mathematical notionof informationpro-gressionintime, andwiththeassociatedcollectionof stochasticprocessescalledmartingales. Wetreat both discrete and continuous timesettings, emphasizing theimportanceofright-continuityofthesamplepathandltrationinthelattercase.Martingalerepresentationsareexplored, aswell asmaximal inequalities, conver-gencetheoremsandapplicationstothestudyofstoppingtimesandtoextinctionofbranchingprocesses.Chapter5providesanintroductiontothebeautiful theoryoftheBrownianmo-tion. ItisrigorouslyconstructedhereviaHilbertspacetheoryandshowntobeaGaussian martingale process of stationary independent increments, with continuoussamplepathandpossessing thestrongMarkov property. Fewofthemanyexplicitcomputationsknownforthisprocessarealsodemonstrated,mostlyinthecontextofhittingtimes,runningmaximaandsamplepathsmoothnessandregularity.56 PREFACEChapter6providesabriefintroductiontothetheoryofMarkovchainsandpro-cesses,avastsubjectatthecoreofprobabilitytheory,towhichmanytextbooksaredevoted. Weillustratesomeoftheinterestingmathematicalpropertiesofsuchprocesses by examining the special case of the Poisson process, and more generally,thatofMarkov jumpprocesses.As clear from the preceding, it normally takes more than a year to cover the scopeof this text. Even more so, given that the intended audience for this course has onlyminimal prior exposure to stochastic processes (beyond the usual elementary prob-abilityclasscoveringonlydiscretesettingsandvariableswithprobabilitydensityfunction). WhilestudentsareassumedtohavetakenarealanalysisclassdealingwithRiemannintegration,nopriorknowledgeofmeasuretheoryisassumedhere.Theunusual solutiontothissetof constraintsistoproviderigorousdenitions,examplesandtheoremstatements, whileforgoingtheproofsof all but themosteasyderivations. Atthissomewhatsuperciallevel,onecancovereverythinginaone semester course of forty lecture hours (and if one has highly motivated studentssuchasIhadinStanford,evenaonequartercourseofthirtylecturehoursmightwork).In preparing thistext Iwas much inuencedby Zakais unpublishedlecturenotes[Zak]. RevisedandexpandedbyShwartzandZeitouni itisusedtothisdayforteachingElectrical EngineeringPhdstudents at theTechnion, Israel. Asecondsource for this text is Breimans [Bre92], which was the intended text book for myclassin2002, till Irealizeditwouldnotdogiventheprecedingconstraints. Theresultingtextisthusamixtureoftheseinuencingfactorswithsomedigressionsandadditionsofmyown.I thankmystudentsoutof whoseworkthistextmaterialized. MostnotablyIthank Nageeb Ali,AjarAshyrkulova, Alessia Falsarone and Che-Lin Su who wrotetherst draft out of notes takeninclass, BarneyHartman-Glaser, Michael He,Chin-LumKwaandChee-HauTanwhousedtheirownclassnotesayearlaterinamajorrevision,reorganization andexpansionofthisdraft,andGaryHuangandMaryTianwhohelpedmewiththeintricaciesof LATEX.I am much indebted to my colleague Kevin Ross for providing many of the exercisesand all the gures in this text. Kevins detailed feedback on an earlier draft of thesenoteshasalsobeenextremelyhelpful inimprovingthepresentationof manykeyconcepts.AmirDemboStanford,CaliforniaJanuary2008CHAPTER1Probability, measureandintegrationThis chapteris devotedtothemathematical foundationsof probabilitytheory.Section1.1introduces the basicmeasuretheoryframework, namely, the proba-bilityspaceandthe-eldsof eventsinit. Thenextbuildingblockarerandomvariables, introduced in Section 1.2 as measurable functions X(). This allowsustodenetheimportantconceptof expectationasthecorrespondingLebesgueintegral,extendingthehorizonofourdiscussionbeyondthespecialfunctionsandvariables withdensity,towhichelementaryprobability theoryislimited. Asmuchofprobabilitytheoryisaboutasymptotics, Section1.3dealswithvariousnotionsof convergenceof randomvariablesandtherelationsbetweenthem. Section1.4concludes the chapter by considering independence and distribution, the two funda-mental aspects that dierentiate probability from (general) measure theory, as wellastherelatedandhighlyuseful technical toolsof weakconvergenceanduniformintegrability.1.1. Probabilityspacesand-eldsWe shall dene here the probability space (, T, P) using the terminology of mea-suretheory. Thesamplespaceisasetof all possibleoutcomes of somerandomexperimentorphenomenon. ProbabilitiesareassignedbyasetfunctionA P(A)toAinasubset Tofallpossiblesetsofoutcomes. Theeventspace Trepresentsboththeamountofinformationavailableasaresultoftheexperimentconductedandthecollectionof all eventsof possibleinterest tous. Apleasantmathematical frameworkresultsbyimposingon Tthestructural conditionsof a-eld,asdoneinSubsection1.1.1. Themostcommonandusefulchoicesforthis-eldarethenexploredinSubsection1.1.2.1.1.1. Theprobabilityspace(, T,P). We use 2to denote theset of allpossiblesubsetsof . Theeventspaceisthusasubset Tof 2, consistingof allallowed events, that is, those events to which we shall assign probabilities. We nextdenethestructuralconditionsimposedon T.Definition1.1.1. Wesaythat T 2isa-eld(ora-algebra), if(a) T,(b)IfA TthenAc Taswell (whereAc= A).(c)IfAi Tfori = 1, 2 . . .thenalso</p> <p>i=1 Ai T.Remark. Using DeMorgans law you can easily check that if Ai Tfor i = 1, 2 . . .and Tisa-eld, thenalso</p> <p>iAi T. Similarly, youcanshowthata-eldisclosedundercountablymanyelementarysetoperations.78 1. PROBABILITY, MEASUREANDINTEGRATIONDefinition1.1.2. Apair (, T) with Ta -eldof subsets of is called ameasurable space. Givena measurable space, a probability measure P is a functionP : T [0, 1],havingthefollowingproperties:(a)0 P(A) 1forall A T.(b)P() = 1.(c) (Countable additivity) P(A) =</p> <p>n=1P(An) whenever A=</p> <p>n=1Anis acountableunionofdisjointsetsAn T(thatis,An</p> <p>Am= ,forall n ,= m).Aprobabilityspaceis atriplet (, T, P), withPaprobabilitymeasureonthemeasurablespace(, T).Thenext exercisecollectssomeof thefundamentalproperties shared byallprob-abilitymeasures.Exercise1.1.3. Let (, T, P) beaprobabilityspaceandA, B, Aievents in T.Provethefollowingpropertiesofeveryprobabilitymeasure.(a) Monotonicity. IfA BthenP(A) P(B).(b) Sub-additivity. IfA iAithenP(A) </p> <p>iP(Ai).(c) Continuityfrombelow: IfAi A,thatis,A1 A2 . . .and iAi= A,thenP(Ai) P(A).(d) Continuityfromabove: IfAi A,thatis,A1 A2 . . .and iAi= A,thenP(Ai) P(A).(e) Inclusion-exclusion rule:P(n_i=1Ai) =</p> <p>iP(Ai) </p> <p>i0foruncountablymanyvaluesof,weshallendupwithP() = . Ofcoursewemaydeneeverythingasbeforeonacountablesubset ofanddemandthatP(A) = P(A)foreachA .Excludingsuchtrivial cases, togenuinelyuseanuncountablesamplespaceweneedtorestrictour-eld Ttoastrictsubsetof2.1.1.2. GeneratedandBorel -elds. Enumeratingthesetsinthe-eldTitnotarealisticoptionforuncountable. Instead, asweseenext, themostcommonconstructionof -eldsisthenbyimplicitmeans. Thatis, wedemandthatcertainsets(calledthegenerators) beinour-eld, andtakethesmallestpossiblecollectionforwhichthisholds.Definition1.1.5. Givenacollectionof subsets A, where anotnecessarilycountableindexset,wedenotethesmallest-eld TsuchthatA Tforall by(A)(orsometimesby(A, )), andcall (A)the-eldgeneratedbythecollection A. Thatis,(A) =</p> <p>(: ( 2isa eld, A ( .Denition1.1.5worksbecausetheintersectionof (possiblyuncountablymany)-eldsisalsoa-eld,whichyouwillverifyinthefollowingexercise.Exercise1.1.6. Let /bea-eldforeach ,anarbitraryindexset. Showthat</p> <p>/isa-eld. Provideanexampleof two-elds Tand (suchthatT (isnota-eld.Dierent sets of generators may result with thesame -eld. For example, taking = 1, 2, 3 it is not hard to check that (1) = (2, 3) = , 1, 2, 3, 1, 2, 3.Example1.1.7. Anexampleof agenerated-eldistheBorel -eldonR. Itmaybedenedas B = ((a, b) : a, b R).Thefollowinglemmalaysoutthestrategyoneemploystoshowthatthe-eldsgenerated bytwodierentcollectionsofsetsareactuallyidentical.Lemma1.1.8. If twodierentcollectionsof generators Aand Baresuchthat A (B) for eachandB (A) for each, then(A) =(B).Proof. Recallthatifacollectionofsets /isasubsetofa-eld (,thenbyDenition1.1.5also(/) (. Applyingthisfor / = Aand (= (B)ourassumption that A (B) for all results with (A) (B). Similarly,our assumptionthat B(A) for all results with(B) (A).Taken together,weseethat(A) = (B).Forinstance, considering BQ=((a, b): a, b Q), wehavebytheprecedinglemmathat BQ= Bassoon asweshow thatanyinterval(a, b)isin BQ. Toverifythisfact,notethat for any real a &lt; bthere are rational numbers qn&lt; rnsuch thatqn aandrn b, hence(a, b)= n(qn, rn) BQ. Followingthesameapproach,you aretoestablishnextafewalternativedenitionsfortheBorel-eld B.Exercise1.1.9. VerifythealternativedenitionsoftheBorel-eld B:((a, b) : a &lt; b R) = ([a, b] : a &lt; b R) = ((, b] : b R)= ((, b] : b Q) = (O R open )10 1. PROBABILITY, MEASUREANDINTEGRATIONHint: AnyO Ropenisacountableunionofsets(a, b)fora, b Q(rational).IfA Risin BofExample1.1.7,wesaythatAisaBorel set. Inparticular,allopenorclosedsubsetsof RareBorelsets,asaremanyothersets. However,Proposition1.1.10. Thereexistsasubsetof Rthatisnotin B. Thatis,notallsetsareBorel sets.Despitetheaboveproposition,allsetsencountered inpracticeareBorelsets.Oftenthereisnoexplicitenumerativedescriptionofthe-eldgeneratedbyaninnitecollectionof subsets. Anotableexceptionis (=([a, b] : a, b Z),where one may check that the sets in (are all possible unions of elements from thecountablecollection b, (b, b + 1), b Z. Inparticular, B ,= (sinceforexample(0, 1/2)/ (.Example1.1.11. Oneexampleof aprobabilitymeasuredenedon(R, B)istheUniformprobabilitymeasureon(0, 1), denotedUanddenedas following. Foreachinterval (a, b) (0, 1), an +n2n1</p> <p>k=0k2n1(k2n,(k+1)2n](x) ,12 1. PROBABILITY, MEASUREANDINTEGRATION0 0.5 101234X()n=10 0.5 101234X()n=20 0.5 101234X()n=30 0.5 101234X()n=4Figure1. Illustration ofapproximation ofarandom variableus-ingsimplefunctionsfordierentvaluesofn.notingthatforR.V.X 0,wehavethatXn= fn(X)aresimplefunctions. SinceX Xn+1 XnandX() Xn() 2nwheneverX() n, itfollowsthatXn() X()asn ,foreach.We write a general R.V. as X() = X+()X() where X+() = max(X(), 0)and X() = min(X(), 0) are non-negative R.V.-s. By theabove argument thesimple functions Xn= fn(X+)fn(X) have the convergence property we claimed.(SeeFigure1foranillustration.)Theconceptofalmostsureprevailsthroughoutprobabilitytheory.Definition1.2.7. Wesaythat R.V. XandY denedonthesameprobabilityspace(, T, P) arealmost surelythesameif P(: X() ,=Y ())=0. Thisshall bedenotedbyXa.s.= Y . Moregenerally, thesamenotationapplies toanyproperty ofaR.V. Forexample,X() 0 a.s. means thatP(: X() &lt; 0) = 0.Hereafter, we shall consider such Xand Yto be the same R.V. hence often omit thequalier a.s.when stating properties of R.V. We also use the terms almost surely(a.s.),almosteverywhere(a.e.),andwithprobability1(w.p.1)interchangeably.The most important -elds are those generated by random variables, as denednext.Definition1.2.8. GivenaR.V. Xwe denoteby (X) the smallest-eld ( Tsuch that X() is measurable on (, (). One can show that (X) = ( : X() ). We call (X) the -eld generated by Xand interchangeably use the notations(X) and TX. Similarly, givenR.V. X1, . . . , Xnonthesamemeasurablespace1.2. RANDOMVARIABLESANDTHEIREXPECTATION 13(, T), denote by (Xk, k n) the smallest -eld Tsuch that Xk(), k = 1, . . . , naremeasurableon(, T). Thatis, (Xk, k n)isthesmallest-eldcontaining(Xk)fork = 1, . . . , n.Remark. One couldalsoconsider the possiblylarger -eld (X) =( :X() B, forall BorelsetsB), butitcanbeshownthat(X)= (X), afactthat we often use in the sequel (that (X) (X) is obvious, and with some eortonecanalsocheckthattheconverseholds).Exercise 1.2.9. Consider a sequence of two coin tosses, = HH, HT, TH, TT,T= 2,= (12). Specify(X0),(X1),and(X2)fortheR.V.-s:X0() = 4,X1() = 2X0()I{1=H}() + 0.5X0()I{1=T}(),X2() = 2X1()I{2=H}() + 0.5X1()I{2=T}().The concept of -eldis neededinorder toproduce arigorous mathematicaltheory. It further hasthecrucial roleof quantifyingtheamount of informa...</p>