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首页 钱钟明+应坚刚.随机分析引论(复旦,英文)

钱钟明+应坚刚.随机分析引论(复旦,英文).pdf

钱钟明+应坚刚.随机分析引论(复旦,英文)

xiu_zhijun
2011-04-17 0人阅读 举报 0 0 暂无简介

简介:本文档为《钱钟明+应坚刚.随机分析引论(复旦,英文)pdf》,可适用于高等教育领域

IntroductiontoStochasticAnalysisZQianandJGYingExeterCollege,OxfordandFudanUniversityJune,iiContentsPreliminariesThemonotoneclasstheoremProbabilitiesandprocessesConditionalexpectationsUniformintegrabilityBorelCantelli’slemmaElementsinthemartingaletheoryMartingalesindiscretetimeDoob’soptionalsamplingtheoremDoob’sinequalitiesThemartingaleconvergencetheoremMartingalesincontinuoustimeLocalmartingalesAdditionaltopicsBrownianmotionConstructionofBrownianmotionScalingpropertiesMarkovpropertyandfinitedimensionaldistributionsThereflectionprincipleMartingalepropertyQuadraticvariationalprocessesAdditionaltopicsItoˆ’scalculusIntroductionQuadraticvariationalprocessesStochasticintegralsforsimpleprocessesStochasticintegralsforadaptedprocessesStochasticintegralsasmartingalesSummaryofmainpropertiesItoˆ’sintegrationforsemimartingalesExtendedtocontinuouslocalmartingalesiiiivCONTENTSExtendedtocontinuoussemimartingalesIto’sformulaItoˆ’sformulaforBMProofofItoˆ’sformulaSelectedapplicationsofItoˆ’sformulaLe´vy’scharacterizationofBrownianmotionTimechangesofBrownianmotionStochasticexponentialsExponentialinequalityGirsanov’stheoremThemartingalerepresentationtheoremAdditionaltopicsStochasticdifferentialequationsIntroductionSeveralexamplesLinearGaussiandiffusionsGeometricBrownianmotionCameronMartin’sformulaStrongsolutions:existenceanduniquenessMartingalesandweaksolutionsAdditionaltopicsMarkovprocessesTransitionsemigroupsKernelsandtheirassociatedoperatorsTransitionfunctionsFellersemigroupsExamplesMarkovpropertySimpleMarkovpropertyRealizationsofMarkovsemigroupsMarkovprocessesintopologicalspacesMarkovpropertyandmartingalepropertyStrongMarkovpropertyFellerprocessesDiffusionprocessesTheWienerspaceDiffusionsasstrongsolutionsofSDE’sAdditionaltopicsAnalysisofMarkovsemigroupsContractionsemigroupsContractionsemigroupsonBanachspacesContractionsemigroupsonHilbertspacesSymmetricMarkovsemigroupsCONTENTSvInvariantandsymmetricmeasuresDirichlet(energy)formsDirichletspacesSymmetricMarkovprocessesEnergyintegralsLyonsZheng’sdecompositionFukushima’sdecompositionPinneddiffusionprocessesConditionaldiffusionsCameronMartin’sformulaforpinneddiffusionsBrownianbridgesAdditionaltopicsAnalysisofDirichletformsHeatsemigroupsRiemannianmetricsTheheatkernelTheheatsemigroupCurvatureanddimensionDiffusionsemigroupsContractivityofdiffusionsemigroupsTheintegralmaximumprincipleUniversalGaussianupperboundHypercontractivityLogarithmicSobolevinequalityOrnsteinUhlenbecksemigroupGross’theoremonhypercontractivityviCONTENTSCONTENTSIntroductionThisbookhasbeenwrittenbasedonthelecturenoteswhichthefirstauthorhasdeliveredtovariousclassesinthelastyearsTheselecturenotesconsistofthematerialwhichbothauthorshavetaughttodifferentlevelgroupsofstudentsatEastChinaNormalUniversity,FudanUniversity,ImperialCollegeLondonandUniversityofOxfordThematerialisparticularlywelcomedbythosepractitionersinstochasticanalysis,stochasticmodellingandmathematicalfinancewhodemandforagoodunderstandingofItoˆ’scalculusandsomecomputationskillsinstochasticcalculusAstochasticdifferentialequationisadifferentialequationperturbedbyrandomnoise(itsintensitymaydependonthetimeparametertandthepositionXtatinstancet),andthereforepossessesaformasfollowingdXtdt=A(t,Xt)σ(t,Xt)W˙tThecentrallimittheoreminprobabilitytheorysuggeststhatW˙tshouldhavenormaldistribution,andforthesackofsimplicity(W˙t)t≥shouldbeindependentatdifferentinstanceSuchrandomnoisemaybemodelledideallybyaBrownianmotion(Wt)t≥,amathematicalmodeldescribingchaoticmovementsofpollenparticlesinaliquid,firstobservedandreportedbyRBrowninAmathematicalformulationofBrownianmotionandthedescriptionofitsdistributionwerederivedbyAlbertEinsteininashortpapertitled“Onthemotionofsmallparticlessuspendedinliquidsatrestrequiredbythemolecularkinetictheoryofheat”publishedin,inAnnalenderPhysik,Aboutthesametime,in,LBacheliersubmittedhisPhDthesisinwhichheusedBrownianmotiontomodelstockmarketsHisresultswerepublishedinapapertitled“The´oriedelaspe´culation”inAnnSciEcoleNormsup,(),,whichisprobablythefirstpaperdevotedtoapplicationsofBrownianmotiontofinanceOntheotherhand,thefirstmathematicalconstructionofBrownianmotionwasachievedin,inthatyearNWienerpublishedhisarticle“Differentialspace”,JMathPhys,FruitfulresultsandmanyunusualfeaturesofBrownianmotionswererevealedmainlybyPaulLe´vyin’s’sAmongofthem,PLe´vyshowedthatalmostsurelyt→Wtisnonwheredifferentiable,andthereforethetimederivativeofBrownianmotion,W˙t,doesnotexistinordinarysenseItisthusnecessarytorewritethepreviousstochasticdifferentialequationindifferentialformdXt=A(t,Xt)dtσ(t,Xt)dWtwhichhastobeinterpretedasanintegralequationXt−X=∫tA(s,Xs)ds∫tσ(s,Xs)dWsCONTENTSItisthusrequiredtodefineintegralslike∫σ(t,Xt)dWtwhichdoesnotexistinordinarysenseItwasKItoˆin’swhofirstestablishedanintegrationtheoryforBrownianmotion,andthereforethetheoryofstochasticdifferentialequationsAmongofthemanifoldapplicationsandconnectionswithpartialdifferentialequations,oneofthemostremarkablerecentapplicationsofItoˆ’stheoryisinthetheoryoffinanceAlthoughItoˆ’stheoryhasnotbeenrecognizedbytheFieldsmedalscommitteeuntiltheawardtoWendelinWernerinforthesophisticatedapplicationstomathematicalphysicsbyhisandhiscoauthors,itwasbroughtworldwiderecognitionbyawardingHMarkowitz,WSharpeandMMillertheNobelPrize,andRobertMerton,andMScholestheNobelPrize,bothinEconomics,butbothforrecognizingtheirworksinvolvingthenovelapplicationsofItoˆ’scalculusintheeconomicsThisbookisamoderateintroductiontothetheoryofstochasticanalysis,andaimstopresentthecorepartofItoˆ’scalculusandthetheoryofMarkovprocessesundertheunifiedmartingaletreatmentThematerialcoveredinthebookprovidesanecessarybackgroundinstochasticanalysisforthosewhoareinterestedinstochasticmodelingsandtheirapplicationsincludingthetheoryoffinance,stochasticcontrolandfilteringetcStudentswhoaremajoringin(pureandapplied)analysis,differentialgeometry,functionalanalysis,harmonicanalysis,mathematicalphysicsandPDEswillfindthisbookrelevanttotheirinterestsZhongminQian,Oxford,UKJiangangYing,Shanghai,ChinaTheworkissupportedpartlybytheNationalBasicResearchProgramofChina(Program:GrantNoCB)ChapterPreliminariesThemonotoneclasstheoremAcollectionLofsubsetsofΩiscalledapisystem,ifitisclosedunderfiniteintersectionsThroughoutthebook,bythemonotoneclasstheorem(respbyamonotoneclassargument),wemeanthefollowinglemmaoraversionofitsvariations(resptheuseofthemonotoneclasstheorem)LemmaLetLbeapisystem,F=σ{L}bethesmallestσalgebrawhichcontainsL,andHbealinearspaceofrealvaluedfunctionsonΩwhichsatisfiestwoconditions:)∈HandA∈HforeveryA∈L)Iffn∈H,fn≥,fnisincreasinginn,andsupnfn<∞,thensupnfn∈HThenHcontainsallbounded,realvaluedandFmeasurablefunctionsonΩProbabilitiesandprocessesLet(Ω,F,P)beaprobabilityspace,whereΩisasamplespace,FisσalgebraonΩ,andPisaprobabilitymeasureon(Ω,F)Ameasurablefunctionon(Ω,F)iscalledarandomvariableintheprobabilitytheoryRecallthatX:Ω→Rdismeasurable,ifforeveryBorelsubsetBofRd,thepreimageX−(B)={ω:X(ω)∈B}belongstoFTherefore,arandomvariableissuchafunctionXonΩthatwemaybeabletotalkabout,forexample,whatistheprobabilityoftheeventthatXliesinaBorelsubsetForp≥LetLp(Ω,F,P)(orsimplyLpifnoconfusionmayarise)betheBanachspaceofpthintegrablerandomvariables,and||X||Lp≡p√E|X|pforeveryX∈Lp(Ω,F,P)Throughoutthisbook,EX(orE(X))denotestheexpectationofrandomvariable,ifitexistsMorepreciselyE(X)≡∫ΩX(ω)P(dω),∀X∈LCHAPTERPRELIMINARIESAsimpleapplicationofHo¨lder’sinequalityshowsthat,ifp≥q,thenLp⊂Lqand||X||Lq≤||X||LpStochasticprocessesaremathematicalmodelstodescriberandomphenomenaevolvingwithtimeInthisbook,weonlystudystochasticprocessesincontinuoustime,thusthesetoftimeparameterwillbe,∞)oritssubset,unlessotherwisespecifiedAstochasticprocessisaparameterizedfamilyX=(Xt)t≥ofrandomvariablestakingvaluesinsometopologicalspaceS,calledthestatespaceofthestochasticprocessXAstochasticprocessX=(Xt)t≥maybeconsideredasafunctionfrom,∞)×Ω→S,whichisthereasonwhyastochasticprocessisalsocalledarandomfunctionAstochasticprocessX=(Xt)t≥isintegrable(resp,squareintegrable)ifeachXtisintegrable(resp,squareintegrable)Foreachsamplepointω∈Ω,functiont→Xt(ω)from,∞)toSiscalledasamplepath(oratrajectory,orasamplefunction)Naturally,astochasticprocessX=(Xt)t≥iscontinuous(resprightcontinuous,rightcontinuouswithleftlimits)ifsamplepathst→Xt(ω)arecontinuous(resprightcontinuous,rightcontinuouswithleftlimits)foralmostallω∈ΩSimilarlyastochasticprocessisboundedifalmostallsamplepathsarecontrolledbyasameconstantExample(ThePoissonprocess)Let(ξn)beasequenceofindependentidenticallydistributed(iid)randomvariableswithexponentialdistributionofparameterλ>LetT=Tn=n∑j=ξjand,foreveryt≥defineXt=nifTn≤t<TnThenforeverysamplepointω,t→Xt(ω)isastepfunction,constantoneach(random)intervalTn,Tn),withjumpat(randomtime)Tn,andisrightcontinuouswithleftlimitn−atTnLetX=(Xt)t≥beastochasticprocesswithstatespaceS,and≤t<t<···<tnThejointdistribution(alsocalledlaw)of(Xt,···,Xtn)isaprobabilitymeasureonS×···×S(withtheproductBorelσalgebra),denotedbyµt,t,···,tn,isdefinedbyµt,t,···,tn(dx,···,dxn)=P(Xt∈dx,···,Xtn∈dxn)µt,t,···,tniscalledafinitedimensionaldistributionofX=(Xt)t≥Moreprecisely,µt,t,···,tnistheuniquemeasureonS×···×Ssuchthatµt,t,···,tn(B×···×Bn)=P(Xt∈B,···,Xtn∈Bn)Therefore,iffisaBorelmeasurablefunctiononS×···×S,then∫S×···×Sf(x,···,xn)µt,t,···,tn(dx,···,dxn)=E(f(Xt,···,Xtn))PROBABILITIESANDPROCESSESprovidedthatf(Xt,···,Xtn)isintegrableIfS=R,ie(Xt)t≥isarealstochasticprocess,thenthefinitedimensionaldistributionµt,t,···,tnisdeterminedbyitsdistributionfunctionFt,t,···,tn(x,···,xn)=P(Xt≤x,···,Xtn≤xn),whereFt,t,···,tn(x,···,xn)=µt,t,···,tn((−∞,x×···×(−∞,xn)Forastochasticprocess(Xt)t≥wearemainlyinterestedinthepropertiesdeterminedbythefamilyofitsfinitedimensionaldistributions{µt,t,···,tn:≤t≤···≤tn}Wemaythereforeconsiderthefamilyoffinitedimensionaldistributionsasthedistributionoftheprocess(Xt)t≥,althoughthelawof(Xt)t≥isindeedaprobabilitymeasureonapathtypespacewhichistypicallyinfinitedimensionalThereisnoadvantagetointroducetheconceptofthelawordistributionforastochasticprocessatthisstage,andwewillavoidthisconceptasfaraspossiblebeforewehavedevelopedItoˆ’scalculusHoweveritseemsquiteessentialtohaveaproperunderstandingoftheconceptofthelawofaprocessinthetheoryofMarkovprocessesTwostochasticprocessesX=(Xt)t≥andY=(Yt)t≥onthesamestatespaceSareequivalentiftheyhavethesamefamilyoffinitedimensionaldistributionsIfX=(Xt)t≥andY=(Yt)t≥areonthesameprobabilityspaceandforeveryt≥P{ω:Xt(ω)=Yt(ω)}=thenwesay(Yt)t≥isaversionof(Xt)t≥(or(Xt)t≥isaversionof(Yt)t≥)Inthiscase,twoprocessesareequivalentTherearetechnicaldifficultieswhenwedealwithstochasticprocessesincontinuoustimeForexample,asubsetofΩlike{ω∈Ω:Xt(ω)∈Bforallt∈,}maybenotmeasurable,ienotevenanevent,sothatP(ω∈Ω:Xt(ω)∈Bforallt∈,)maynotmakesense,unlessadditionalconditionson(Xt)t≥areimposedSimilarly,afunctionlikesupt∈KXtmaybenotmeasurableSuchsituationwillbeveryinconvenientToavoidsuchtechnicaldifficulties,acommoncondition,whichisgoodenoughtoincludealargeclassofinterestingstochasticprocesses,isthatXisrightcontinuousalmostsurely,and(Ω,F,P)iscompleteinthesensethatanytrivialsubsetsofprobabilityareeventsExerciseLet(Xt)t≥beastochasticprocessinRdon(Ω,F,P),andBbeaBorelmeasurablesubsetIfFisafiniteorcountablesubsetof,∞),thenboth{ω:Xt(ω)∈Bforanyt∈F}andsupt∈F|Xt|aremeasurableCHAPTERPRELIMINARIESInpracticalsituations,wearegivenacollectionofconsistentfinitedimensionaldistributionsD={µt,···,tn,fort<···<tn,tj∈,∞)},wewouldliketoconstructastochasticprocess(Xt)t≥onsomeprobabilityspace(Ω,F,P)sothatthefamilyoffinitedimensionaldistributionsdeterminedby(Xt)t≥isthegivenfamilyDInthiscase,(Xt)t≥iscalledarealizationofDConditionalexpectationsThemainconceptsintheprobabilitytheory,includingindependence,martingalepropertyandMarkovproperty,arestatedintermsofconditionalexpectations(andconditionalprobability)LetXbeanintegrablerandomvariableon(Ω,F,P),andGbeasubσalgebraofFThenthereisauniqueintegrablerandomvariable,denotedbyE(X|G),calledtheconditionalexpectationofXgivenG,suchthat)E(X|G)ismeasurablewithrespecttoG,and)ForanyA∈GEE(X|G)A=E(XA)TheconditionalexpectationE(X|G)isthebestprediction(underthemeansquaredistance)oftherandomvariableXbasedonavailableinformationGAccordingto),)andthemonotoneclasstheoremE(E(X|G)Y)=E(XY)providedthatYisGmeasurableandXYisintegrableIfXandYaretworandomvariablesandXisintegrable,thenE(X|Y)meansE(X|σ(Y)),whereσ(Y)isthesmallestσalgebraforwhichYismeasurableItcanbeshownthatE(X|Y)isameasurablefunctionofY,thatis,thereisaBorelfunctionFsuchthatE(X|Y)=F(Y)ExerciseIfξ,ηaretworandomvariablesandξisσ(η)measurable,thenξ=f(η)forsomeBorelfunctionfAccordingtodefinition,ifYisGmeasurable,thenE(YX|G)=YE(X|G)IfXandGareindependent,thenE(X|G)

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钱钟明+应坚刚.随机分析引论(复旦,英文)

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