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首页 Klebaner.随机分析导论及其应用

Klebaner.随机分析导论及其应用.pdf

Klebaner.随机分析导论及其应用

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

简介:本文档为《Klebaner.随机分析导论及其应用pdf》,可适用于高等教育领域

INTRODUCTIONTOSTOCHASTICCALCULUSWITHAPPLICATIONSSECONDEDITIONThispageintentionallyleftblankFimaCKlebanerMonashUniversity,AustraliaImperialCollegePressBritishLibraryCataloguinginPublicationDataAcataloguerecordforthisbookisavailablefromtheBritishLibraryPublishedbyImperialCollegePressSheltonStreetCoventGardenLondonWCHHEDistributedbyWorldScientificPublishingCoPteLtdTohTuckLink,SingaporeUSAoffice:WarrenStreet,Suite,Hackensack,NJUKoffice:SheltonStreet,CoventGarden,LondonWCHHEPrintedinSingaporeForphotocopyingofmaterialinthisvolume,pleasepayacopyingfeethroughtheCopyrightClearanceCenter,Inc,RosewoodDrive,Danvers,MA,USAInthiscasepermissiontophotocopyisnotrequiredfromthepublisherISBNISBNX(pbk)AllrightsreservedThisbook,orpartsthereof,maynotbereproducedinanyformorbyanymeans,electronicormechanical,includingphotocopying,recordingoranyinformationstorageandretrievalsystemnowknownortobeinvented,withoutwrittenpermissionfromthePublisherCopyright©byImperialCollegePressINTRODUCTIONTOSTOCHASTICCALCULUSWITHAPPLICATIONS(SecondEdition)PrefacePrefacetotheSecondEditionThesecondeditionisrevised,expandedandenhancedThisisnowamorecompletetextinStochasticCalculus,frombothatheoreticalandanapplicationspointofviewChangescameabout,asaresultofusingthisbookforteachingcoursesinStochasticCalculusandFinancialMathematicsoveranumberofyearsManytopicsareexpandedwithmoreworkedoutexamplesandexercisesSolutionstoselectedexercisesareincludedAnewchapteronbondsandinterestratescontainsderivationsofthemainpricingmodels,includingcurrentlyusedmarketmodels(BGM)Thechangeofnumerairetechniqueisdemonstratedoninterestrate,currencyandexoticoptionsThepresentationofApplicationsinFinanceisnowmorecomprehensiveandselfcontainedThemodelsinBiologyintroducedintheneweditionincludetheagedependentbranchingprocessandastochasticmodelforcompetitionofspeciesTheseMarkovprocessesaretreatedbyStochasticCalculustechniquesusingsomenewrepresentations,suchasarelationbetweenPoissonandBirthDeathprocessesThemathematicaltheoryoffilteringisbasedonthemethodsofStochasticCalculusInthenewedition,wederivestochasticequationsforanonlinearfilterfirstandobtaintheKalmanBucyfilterasacorollaryModelsarisinginapplicationsaretreatedrigorouslydemonstratinghowtoapplytheoreticalresultstoparticularmodelsThisapproachmightnotmakecertainplaceseasyreading,however,byusingthisbook,thereaderwillaccomplishaworkingknowledgeofStochasticCalculusPrefacetotheFirstEditionThisbookaimsatprovidingaconcisepresentationofStochasticCalculuswithsomeofitsapplicationsinFinance,EngineeringandScienceDuringthepasttwentyyears,therehasbeenanincreasingdemandfortoolsandmethodsofStochasticCalculusinvariousdisciplinesOneofthegreatestdemandshascomefromthegrowingareaofMathematicalFinance,whereStochasticCalculusisusedforpricingandhedgingoffinancialderivatives,vviPREFACEsuchasoptionsInEngineering,StochasticCalculusisusedinfilteringandcontroltheoryInPhysics,StochasticCalculusisusedtostudytheeffectsofrandomexcitationsonvariousphysicalphenomenaInBiology,StochasticCalculusisusedtomodeltheeffectsofstochasticvariabilityinreproductionandenvironmentonpopulationsFromanappliedperspective,StochasticCalculuscanbelooselydescribedasafieldofMathematics,thatisconcernedwithinfinitesimalcalculusonnondifferentiablefunctionsTheneedforthiscalculuscomesfromthenecessitytoincludeunpredictablefactorsintomodellingThisiswhereprobabilitycomesinandtheresultisacalculusforrandomfunctionsorstochasticprocessesThisisamathematicaltext,thatbuildsontheoryoffunctionsandprobabilityanddevelopsthemartingaletheory,whichishighlytechnicalThistextisaimedatgraduallytakingthereaderfromafairlylowtechnicalleveltoasophisticatedoneThisisachievedbymakinguseofmanysolvedexamplesEveryefforthasbeenmadetokeeppresentationassimpleaspossible,whilemathematicallyrigorousSimpleproofsarepresented,butmoretechnicalproofsareleftoutandreplacedbyheuristicargumentswithreferencestoothermorecompletetextsThisallowsthereadertoarriveatadvancedresultssoonerTheseresultsarerequiredinapplicationsForexample,thechangeofmeasuretechniqueisneededinoptionspricingcalculationsofconditionalexpectationswithrespecttoanewfiltrationisneededinfilteringItturnsoutthatcompletelyunrelatedappliedproblemshavetheirsolutionsrootedinthesamemathematicalresultForexample,theproblemofpricinganoptionandtheproblemofoptimalfilteringofanoisysignal,bothrelyonthemartingalerepresentationpropertyofBrownianmotionThistextpresumeslessinitialknowledgethanmosttextsonthesubject(Me´tivier(),DellacherieandMeyer(),Protter(),LiptserandShiryayev(),JacodandShiryayev(),KaratzasandShreve(),StroockandVaradhan(),RevuzandYor(),RogersandWilliams()),howeveritstillpresentsafairlycompleteandmathematicallyrigoroustreatmentofStochasticCalculusforbothcontinuousprocessesandprocesseswithjumpsAbriefdescriptionofthecontentsfollows(formoredetailsseetheTableofContents)ThefirsttwochaptersdescribethebasicresultsinCalculusandProbabilityneededforfurtherdevelopmentThesechaptershaveexamplesbutnoexercisesSomemoretechnicalresultsinthesechaptersmaybeskippedandreferredtolaterwhenneededInChapter,thetwomainstochasticprocessesusedinStochasticCalculusaregiven:Brownianmotion(forcalculusofcontinuousprocesses)andPoissonprocess(forcalculusofprocesseswithjumps)IntegrationwithrespecttoBrownianmotionandcloselyrelatedprocesses(Itoˆprocesses)isintroducedinChapterItallowsonetodefineastochasticdifferentialequationSuchPREFACEviiequationsariseinapplicationswhenrandomnoiseisintroducedintoordinarydifferentialequationsStochasticdifferentialequationsaretreatedinChapterDiffusionprocessesariseassolutionstostochasticdifferentialequations,theyarepresentedinChapterAsthenamesuggests,diffusionsdescribearealphysicalphenomenon,andaremetinmanyreallifeapplicationsChaptercontainsinformationaboutmartingales,examplesofwhichareprovidedbyItoˆprocessesandcompensatedPoissonprocesses,introducedinearlierchaptersThemartingaletheoryprovidesthemaintoolsofstochasticcalculusTheseincludeoptionalstopping,localizationandmartingalerepresentationsTheseareabstractconcepts,buttheyariseinappliedproblems,wheretheiruseisdemonstratedChaptergivesabriefaccountofcalculusformostgeneralprocesses,calledsemimartingalesBasicresultsincludeItoˆ’sformulaandstochasticexponentialThereaderhasalreadymettheseconceptsinBrownianmotioncalculusgiveninChapterChaptertreatsPureJumpprocesses,wheretheyareanalyzedbyusingcompensatorsThechangeofmeasureisgiveninChapterThistopicisimportantinoptionspricing,andforinferenceforstochasticprocessesChaptersaredevotedtoapplicationsofStochasticCalculusApplicationsinFinancearegiveninChaptersand,stocksandcurrencyoptions(Chapter)bonds,interestratesandtheiroptions(Chapter)ApplicationsinBiologyaregiveninChapterTheyincludediffusionmodels,BirthDeathprocesses,agedependent(BellmanHarris)branchingprocesses,andastochasticversionoftheLotkaVolterramodelforcompetitionofspeciesChaptergivesapplicationsinEngineeringandPhysicsEquationsforanonlinearfilterarederived,andappliedtoobtaintheKalmanBucyfilterRandomperturbationstotwodimensionaldifferentialequationsaregivenasanapplicationinPhysicsExercisesareplacedattheendofeachchapterThistextcanbeusedforavarietyofcoursesinStochasticCalculusandFinancialMathematicsTheapplicationtoFinanceisextensiveenoughtouseitforacourseinMathematicalFinanceandforselfstudyThistextissuitableforadvancedundergraduatestudents,graduatestudentsaswellasresearchworkersandpractionersAcknowledgmentsThankstoRobertLiptserandKaisHamzawhoprovidedmostvaluablecommentsThankstotheEditorLenoreBettsforproofreadingthendeditionTheremainingerrorsaremyownThankstomycolleaguesandstudentsfromuniversitiesandbanksThankstomyfamilyforbeingsupportiveandunderstandingFimaCKlebanerMonashUniversityMelbourne,ThispageintentionallyleftblankContentsPrefacevPreliminariesFromCalculusFunctionsinCalculusVariationofaFunctionRiemannIntegralandStieltjesIntegralLebesgue’sMethodofIntegrationDifferentialsandIntegralsTaylor’sFormulaandOtherResultsConceptsofProbabilityTheoryDiscreteProbabilityModelContinuousProbabilityModelExpectationandLebesgueIntegralTransformsandConvergenceIndependenceandCovarianceNormal(Gaussian)DistributionsConditionalExpectationStochasticProcessesinContinuousTimeBasicStochasticProcessesBrownianMotionPropertiesofBrownianMotionPathsThreeMartingalesofBrownianMotionMarkovPropertyofBrownianMotionHittingTimesandExitTimesMaximumandMinimumofBrownianMotionDistributionofHittingTimesReflectionPrincipleandJointDistributionsZerosofBrownianMotionArcsineLawixxPREFACESizeofIncrementsofBrownianMotionBrownianMotioninHigherDimensionsRandomWalkStochasticIntegralinDiscreteTimePoissonProcessExercisesBrownianMotionCalculusDefinitionofItoˆIntegralItoˆIntegralProcessItoˆIntegralandGaussianProcessesItoˆ’sFormulaforBrownianMotionItoˆProcessesandStochasticDifferentialsItoˆ’sFormulaforItoˆProcessesItoˆProcessesinHigherDimensionsExercisesStochasticDifferentialEquationsDefinitionofStochasticDifferentialEquationsStochasticExponentialandLogarithmSolutionstoLinearSDEsExistenceandUniquenessofStrongSolutionsMarkovPropertyofSolutionsWeakSolutionstoSDEsConstructionofWeakSolutionsBackwardandForwardEquationsStratanovichStochasticCalculusExercisesDiffusionProcessesMartingalesandDynkin’sFormulaCalculationofExpectationsandPDEsTimeHomogeneousDiffusionsExitTimesfromanIntervalRepresentationofSolutionsofODEsExplosionRecurrenceandTransienceDiffusiononanIntervalStationaryDistributionsMultiDimensionalSDEsExercisesPREFACExiMartingalesDefinitionsUniformIntegrabilityMartingaleConvergenceOptionalStoppingLocalizationandLocalMartingalesQuadraticVariationofMartingalesMartingaleInequalitiesContinuousMartingalesChangeofTimeExercisesCalculusForSemimartingalesSemimartingalesPredictableProcessesDoobMeyerDecompositionIntegralswithrespecttoSemimartingalesQuadraticVariationandCovariationItoˆ’sFormulaforContinuousSemimartingalesLocalTimesStochasticExponentialCompensatorsandSharpBracketProcessItoˆ’sFormulaforSemimartingalesStochasticExponentialandLogarithmMartingale(Predictable)RepresentationsElementsoftheGeneralTheoryRandomMeasuresandCanonicalDecompositionExercisesPureJumpProcessesDefinitionsPureJumpProcessFiltrationItoˆ’sFormulaforProcessesofFiniteVariationCountingProcessesMarkovJumpProcessesStochasticEquationforJumpProcessesExplosionsinMarkovJumpProcessesExercisesxiiPREFACEChangeofProbabilityMeasureChangeofMeasureforRandomVariablesChangeofMeasureonaGeneralSpaceChangeofMeasureforProcessesChangeofWienerMeasureChangeofMeasureforPointProcessesLikelihoodFunctionsExercisesApplicationsinFinance:StockandFXOptionsFinancialDerivativesandArbitrageAFiniteMarketModelSemimartingaleMarketModelDiffusionandtheBlackScholesModelChangeofNumeraireCurrency(FX)OptionsAsian,LookbackandBarrierOptionsExercisesApplicationsinFinance:Bonds,RatesandOptionsBondsandtheYieldCurveModelsAdaptedtoBrownianMotionModelsBasedontheSpotRateMerton’sModelandVasicek’sModelHeathJarrowMorton(HJM)ModelForwardMeasuresBondasaNumeraireOptions,CapsandFloorsBraceGatarekMusiela(BGM)ModelSwapsandSwaptionsExercisesApplicationsinBiologyFeller’sBranchingDiffusionWrightFisherDiffusionBirthDeathProcessesBranchingProcessesStochasticLotkaVolterraModelExercisesPREFACExiiiApplicationsinEngineeringandPhysicsFilteringRandomOscillatorsExercisesSolutionstoSelectedExercisesReferencesIndexThispageintentionallyleftblankChapterPreliminariesFromCalculusStochasticcalculusdealswithfunctionsoftimet,≤t≤TInthischaptersomeconceptsoftheinfinitesimalcalculususedinthesequelaregivenFunctionsinCalculusContinuousandDifferentiableFunctionsAfunctiongiscalledcontinuousatthepointt=tiftheincrementofgoversmallintervalsissmall,∆g(t)=g(t)−g(t)→as∆t=t−t→Ifgiscontinuousateverypointofitsdomainofdefinition,itissimplycalledcontinuousgiscalleddifferentiableatthepointt=tifatthatpoint∆g∼C∆torlim∆t→∆g(t)∆t=C,thisconstantCisdenotedbyg′(t)Ifgisdifferentiableateverypointofitsdomain,itiscalleddifferentiable

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