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首页 Pham.连续时间随机控制与优化及其在金融中的应用

Pham.连续时间随机控制与优化及其在金融中的应用.pdf

Pham.连续时间随机控制与优化及其在金融中的应用

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

简介:本文档为《Pham.连续时间随机控制与优化及其在金融中的应用pdf》,可适用于高等教育领域

StochasticMechanicsRandomMediaSignalProcessingandImageSynthesisMathematicalEconomicsandFinanceStochasticOptimizationStochasticControlStochasticModelsinLifeSciencesStochasticModellingandAppliedProbability(Formerly:ApplicationsofMathematics)EditedbyBRozovskiı˘GGrimmettAdvisoryBoardDDawsonDGemanIKaratzasFKellyYLeJanBØksendalGPapanicolaouEPardouxHuyênPhamContinuoustimeStochasticControlandOptimizationwithFinancialApplicationsHuyênPhamUniversitéParisDenisDiderotUFRMathématiquesSiteChevaleret,CaseParisCedexFrancephammathjussieufrManagingEditorsBorisRozovskiı˘DivisionofAppliedMathematicsBrownUniversityGeorgeStProvidence,RIUSArozovskydambrowneduGeoffreyGrimmettCentreforMathematicalSciencesUniversityofCambridgeWilberforceRoadCambridgeCBWBUKgrgrimmettstatslabcamacukISSNISBNeISBNDOISpringerDordrechtHeidelbergLondonNewYorkLibraryofCongressControlNumber:MathematicsSubjectClassification():E,B,L,L,Hc©SpringerVerlagBerlinHeidelbergThisworkissubjecttocopyrightAllrightsarereserved,whetherthewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting,reproductiononmicrofilmorinanyotherway,andstorageindatabanksDuplicationofthispublicationorpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember,,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringerViolationsareliabletoprosecutionundertheGermanCopyrightLawTheuseofgeneraldescriptivenames,registerednames,trademarks,etcinthispublicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluseCoverdesign:deblikPrintedonacidfreepaperSpringerispartofSpringerScienceBusinessMedia(wwwspringercom)ToChaˆu,HugoandAntoinePrefaceDynamicstochasticoptimizationisthestudyofdynamicalsystemssubjecttorandomperturbations,andwhichcanbecontrolledinordertooptimizesomeperformancecriterionItarisesindecisionmakingproblemsunderuncertainty,andfindsnumerousandvariousapplicationsineconomics,managementandfinanceHistoricallyhandledwithBellman’sandPontryagin’soptimalityprinciples,theresearchoncontroltheoryhasconsiderablydevelopedoverrecentyears,inspiredinparticularbyproblemsemergingfrommathematicalfinanceThedynamicprogrammingprinciple(DPP)toastochasticcontrolproblemforMarkovprocessesincontinuoustimeleadstoanonlinearpartialdifferentialequation(PDE),calledtheHamiltonJacobiBellman(HJB)equation,satisfiedbythevaluefunctionTheglobalapproachforstudyingstochasticcontrolproblemsbytheBellmanDPPhasasuitableframeworkinviscositysolutions,whichhavebecomepopularinmathematicalfinance:thisallowsustogobeyondtheclassicalverificationapproachbyrelaxingthelackofregularityofthevaluefunction,andbydealingwithdegeneratesingularcontrolsproblemsarisingtypicallyinfinanceThestochasticmaximumprinciplefoundamodernpresentationwiththeconceptofbackwardstochasticdifferentialequations(BSDEs),whichledtoaveryactiveresearchareawithinterestingapplicationsinstochasticanalysis,PDEtheoryandmathematicalfinanceOntheotherhand,andmotivatedbyportfoliooptimizationproblemsinfinance,anotherapproach,calledtheconvexdualitymartingalemethod,developedandgeneratedanimportantliteratureItreliesonrecentresultsinstochasticanalysisandonclassicalmethodsinconvexanalysisandoptimizationThereexistseveralmonographsdealingwitheitherthedynamicprogrammingapproachforstochasticcontrolproblems(FR,BL,Kry,FSo,YZ,T)orbackwardstochasticdifferentialequations(ElkM,MY)Theymainlyfocusonthetheoreticalaspects,andaretechnicallyofadvancedlevel,andusuallydifficulttoreadforanonexpertinthetopicMoreover,althoughtherearemanypapersaboututilitymaximizationbydualitymethods,thisapproachisrarelyaddressedingraduateandresearchbooks,withtheexceptionoftheforthcomingoneFBSThepurposeofthisbookistofillinthisgap,andtoprovideasystematictreatmentofthedifferentaspectsintheresolutionofstochasticoptimizationproblemsincontinuoustimewithaviewtowardsfinancialapplicationsWeincludedrecentdevelopmentsandoriginalresultsonthisfield,whichappearinmonographformforthefirsttimeWepaidVIIIPrefaceattentiontothepresentationofanaccessibleversionofthetheoryforthosewhoarenotnecessarilyexpertsonstochasticcontrolAlthoughtheresultsarestatedinarathergeneralframework,usefulforthevariousapplications,withcompleteanddetailedproofs,wehaveoutlinedtheintuitionbehindsomeadvancedmathematicalconceptsWealsotakecaretoillustrateeachoftheresolutionmethodologiesusingseveralexamplesinfinanceThismonographisdirectedtowardsgraduatestudentsandresearchersinmathematicalfinanceItwillalsoappealtoappliedmathematiciansinterestedinfinancialapplicationsandpractitionerswishingtoknowmoreabouttheuseofstochasticoptimizationmethodsinfinanceThebookisorganizedasfollowsSinceitisintendedtobeselfcontained,westartbyrecallinginChaptersomeprerequesitesinstochasticcalculusWeessentiallycollectnotionsandresultsinstochasticanalysisthatwillbeusedinthefollowingchaptersandmayalsoserveasaquickreferenceforknowledgeablereadersInChapter,weformulateingeneraltermsthestructureofastochasticoptimizationproblem,andoutlineseveralexamplesofrealapplicationsineconomicsandfinanceAnalysisandsolutionstotheseexampleswillbedetailedinthesubsequentchaptersbydifferentapproachesWealsobrieflydiscussothercontrolmodelsthantheonestudiedinthisbookChapterpresentsthedynamicprogrammingmethodforcontrolleddiffusionprocessesTheclassicalapproachbasedonaverificationtheoremfortheHJBequationwhenthevaluefunctionissmooth,isdetailedandillustratedinvariousexamples,includingthestandardMertonportfolioselectionproblemInChapter,weadopttheviscositysolutionsapproachfordynamicprogrammingequationstostochasticcontrolproblemsThisavoidstheaprioriassumptionofsmoothnessofthevaluefunction,whichisdesirableasitisoftennotsmoothSomeoriginalproofsaredetailedinaunifyingframeworkembeddingbothregularandsingularcontrolproblemsAsectionisdevotedtocomparisonprinciples,whicharekeypropertiesinviscositysolutionstheory,astheyprovideuniquecharacterizationofthevaluefunctionIllustrativeexamplescomingfromfinancecompletethischapterInChapter,weconsideroptimalstoppingandswitchingcontrolproblems,whichconstituteaclassicalandimportantclassofstochasticcontrolproblemsTheseproblemshaveattractedanincreasinglyrenewedinterestduetotheirvariousapplicationsinfinanceWerevisittheirtreatmentbymeansofviscositysolutionstotheassociateddynamicprogrammingfreeboundaryproblemsWegiveexplicitsolutionstoseveralexamplesarisingfromtherealoptionsliteratureAsmentionedabove,thePontryaginmaximumprincipleleadsnaturallytothenotionofbackwardstochasticdifferentialequationsChapterisanintroductiontothistheory,insistingespeciallyontheapplicationsofBSDEstostochasticcontrol,andtoitsrelationwithnonlinearPDEsthroughFeynmanKactypeformulaeWealsoconsiderreflectedBSDEs,whicharerelatedtooptimalstoppingproblemsandvariationalinequalitiesTwoapplicationsinoptionhedgingproblemsaresolvedbytheBSDEmethodInChapter,wepresenttheconvexdualitymartingaleapproachthatoriginatesfromportfoliooptimizationproblemThestartingpointofthismethodisadualrepresentationforthesuperreplicationcostofoptionsrelyingonpowerfuldecompositiontheoremsinstochasticanalysisWethenstateageneralexistenceandcharacterizationresultfortheutilitymaximizationproblembydualitymethods,andillustrateinsomeparticularexampleshowitleadstoexplicitsolutionsWealsoconsiderthepopularmeanvariancehedgingproblemthatwestudybyadualityapproachPrefaceIXThisbookisbasedmainlyonmyresearchstudies,andalsoonlecturenotesforgraduatecoursesintheMaster’sprogramsofmathematicalfinanceatUniversitiesParisandParisPartofitwasalsousedasmaterialforanoptionalcourseatENSAEinMalakoffThiseditionisanupdatedandexpandedversionofmybookpublishedinFrenchbySpringerinthecollectionMathe´matiquesetApplicationsoftheSMAIThetextiswidelyreworkedtotakeintoaccounttherapidevolutionofsomeofthesubjectstreatedAnewChapteronoptimalswitchingproblemshasbeenaddedChapterontheviscositysolutionsapproachhasbeenlargelyrewritten,withadetailedtreatmentoftheterminalcondition,andofcomparisonprinciplesWealsoincludedinChapteranewsectiononreflectedBSDEs,whicharerelatedtooptimalstoppingproblems,andgenerateaveryactiveresearchareaIwishtothankNicoleElKaroui,whosubstantiallyreviewedseveralchapters,andmadehelpfulcommentsHerseminalworksonstochasticcontrolandmathematicalfinanceprovidedarichsourceforthisbookSeveralexpertsandfriendshaveshowntheirinterestandsupport:BrunoBouchard,RamaCont,MoniqueJeanblanc,DamienLamberton,PhilipProtter,DenisTalayandNizarTouziIamgratefultoMoniquePontier,whoreviewedtheFrencheditionofthisbookforMathSciNet,andpointedoutseveralmisprintswithusefulremarksLastbutnotleast,IwouldliketothankChaˆu,HugoandAntoineforalltheirloveParis,DecemberHuyeˆnPHAMContentsSomeelementsofstochasticanalysisStochasticprocessesFiltrationandprocessesStoppingtimesBrownianmotionMartingales,semimartingalesStochasticintegralandapplicationsStochasticintegralwithrespecttoacontinuoussemimartingaleItoˆprocessItoˆ’sformulaMartingalerepresentationtheoremGirsanov’stheoremStochasticdifferentialequationsStrongsolutionsofSDEEstimatesonthemomentsofsolutionstoSDEFeynmanKacformulaStochasticoptimizationproblemsExamplesinfinanceIntroductionExamplesPortfolioallocationProductionconsumptionmodelIrreversibleinvestmentmodelQuadratichedgingofoptionsSuperreplicationcostinuncertainvolatilityOptimalsellingofanassetValuationofnaturalresourcesOtheroptimizationproblemsinfinanceErgodicandrisksensitivecontrolproblemsSuperreplicationundergammaconstraintsXIIContentsRobustutilitymaximizationproblemandriskmeasuresForwardperformancecriterionBibliographicalremarksTheclassicalPDEapproachtodynamicprogrammingIntroductionControlleddiffusionprocessesDynamicprogrammingprincipleHamiltonJacobiBellmanequationFormalderivationofHJBRemarksandextensionsVerificationtheoremApplicationsMertonportfolioallocationprobleminfinitehorizonInvestmentconsumptionproblemwithrandomtimehorizonAmodelofproductionconsumptiononinfinitehorizonExampleofsingularstochasticcontrolproblemBibliographicalremarksTheviscositysolutionsapproachtostochasticcontrolproblemsIntroductionDefinitionofviscositysolutionsFromdynamicprogrammingtoviscositysolutionsofHJBequationsViscositypropertiesinsidethedomainTerminalconditionComparisonprinciplesanduniquenessresultsClassicalcomparisonprincipleStrongcomparisonprincipleAnirreversibleinvestmentmodelProblemRegularityandconstructionofthevaluefunctionOptimalstrategySuperreplicationcostinuncertainvolatilitymodelBoundedvolatilityUnboundedvolatilityBibliographicalremarksOptimalswitchingandfreeboundaryproblemsIntroductionOptimalstoppingDynamicprogrammingandviscositypropertySmoothfitprincipleOptimalstrategyMethodsofsolutionintheonedimensionalcaseExamplesofapplicationsContentsXIIIOptimalswitchingProblemformulationDynamicprogrammingandsystemofvariationalinequalitiesSwitchingregionsTheonedimensionalcaseExplicitsolutioninthetworegimecaseBibliographicalremarksBackwardstochasticdifferentialequationsandoptimalcontrolIntroductionGeneralpropertiesExistenceanduniquenessresultsLinearBSDEComparisonprinciplesBSDE,PDEandnonlinearFeynmanKacformulaeControlandBSDEOptimizationofafamilyofBSDEsStochasticmaximumprincipleReflectedBSDEsandoptimalstoppingproblemsExistenceandapproximationviapenalizationConnectionwithvariationalinequalities

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Pham.连续时间随机控制与优化及其在金融中的应用

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