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Continuous-TimeOptimalManagementforthePensionFunds::::20125:::::2040CIRVasicekHJBLegendre-CEVGBMCEV-Legendre-CEVi5.1GBMnqi2[¡1;1]-RiccatiHJBCEV5.2´LegendreCEVHestonHestonHJBCEVHestonLegendreHamilton-Jacobi-Bellman(HJB)iiABSTRACTBythedevelopmentoftheworldeconomyandthesociety,thepensioninsurancesystemplaysamoreandmoreimportantroleforthebasiclivelihoodoftheelderly.Underthebackgroundofthewholeworldoldage,thepropor-tionandnumberoftheelderlyarebothmuchhigher.Thepensioninsuranceensuresthebasicneedsinthelifeoftheelderly,whichmeansguaranteethebasiclifeofthequitepartialpopulation.Therefore,thepensioninsurancehasbecomeafocusofglobalattention,andithasaveryimportantroleforthesocialstabilityanddevelopment.Asthepopulationevolutionandeconomic°uctuation,thepensioninsurancefacesmanyuncertainfactors,suchasthesalarystructure,theworker°ow,thestockprice°uctuation,in°ationandin-terestratechangeandsoon.Asisknowntoall,thesedemographicvariablesandeconomicvariablesarenotcertain,butarerandom.Thus,therandomsimulationandrandomcontroltheoryhavealreadybeenusedintheoptimalmanagementofthepensioninsurance.Bythemathematicalmethod,thispa-perresearchestheoptimalmanagementofde¯ned-contributionpensionfundsunderthecontinuoustimeinordertoenrichthetheoryandpractice.Becausethepensioninvestmentisalong-terminvestment,generally20to40years,insuchalongtime,theinterestratesarerandom°uctuation;thepensioninvestmentmustconsidertheinterestraterisk.Chapter3studiestheoptimalinvestmentofde¯ned-contributionpensionunderthea±neinterestratemodel(includingCIRmodelandVasicekmodel).Inthemodel,thepensionfundsareallowedtoinvestinarisk-freeasset,azerocouponbondandariskasset.BytheapplicationofHJBequation,Legendreconversionanddualtheory,we¯ndtheexplicitsolutionoftheutilityfunctionsofconstantrelativeriskaversionandabsoluteriskaversion.Theresearchconclusionshave:inthepowerutilityfunction,alongwiththeretirementisnear,thehigh-riskstockinvestmentbegangraduallytoturntothelow-riskcashoriiibondinvestments,butthecashandbondinvestmenttrendareuncertain;intheexponentialutilityfunction,withtheapproachoftheretiredmoment,thefundmanagerwillinvestmoreinhigh-riskstock,andlessinlow-riskcashandbonds,butcan'talsodeterminethelow-riskassetsallocationproportion.Finally,makethenumericalanalysis.Basedonthede¯cienciesoftheutilitymaximization,Chapter4adoptsthemean-variancemodeltoknowthedecision-makingprocessoftheweightoftherisksandbene¯tsbetter.Becausetheconstantvarianceelasticity(CEV)modelisoftenthenaturalextensionofgeometricBrownianmotion(GBM),theriskyassetspriceismorepracticalwhichobeystheCEVmodel.Bytherandomcontroltheory,establishestheoptimalinvestmentproblemofthepen-sionfundsunderthemean-variancemodel.BytheLegendretransformationandthedualtheory,obtainstheoptimalinvestmentstrategiesbeforeandafterretirement,anddevisesthee®ectivefrontierunderthemean-variancemodel.AlthoughChapter4assumesthattheriskyassetspriceobeystheCEVmodel,the¯nancialmarketonlycontainsariskyassetanddoesn'tcon-siderthestochasticsalary.Theseassumptionsarestricter,soSection5.1assumesthatthenkindsofriskyassetspricesallobeystheGBMmodel,andthereisacorrelationcoe±cientqi2[¡1;1]betweentherisky-assetsstochasticsourcesandthesalarystochasticsources.Moreover,themean-variancemodelisstilltheoptimizingtarget.ByconstructingaRiccatiequationastheHJBequation'ssolution,ultimatelyobtaintheexplicitsolutionandthee®ectivefrontier.Theabovestudiesdon'tallconsidercombiningtheCEVmodelandthestochasticsalary,soSection5.2establishesthemathematicalmodelsunderthelogarithmicutilityandutility,respectively,andobtainstheexplicitsolutionsbytheapplicationofstochasticdynamicprogrammingtechniques.Assumingthatthevolatilityofthesalarycompletelycomesfromthe¯nancialmarket,thereisthemultiplerelationship(´),whichmeasuringhowtherisksourcesofstocka®ectthesalary.Inthelogarithmicutilityfunction,studiestheoptimalivinvestmentproblemofthepensionfundswiththesalaryasabenchmark,andobtainstheoptimalinvestmentproportionbythestochasticcontrol,Legendreconversionanddualitytheory.Intheexponentialutilityfunction,throughthepowertransformationandvariablereplacement,convertsthenonlinearequationsintolinearones,andultimatelyobtainstheexplicitsolutionbyconstructingthesolution.Finally,theCEVmodelisthepricedependentvolatilitymodel,whosevolatilityisn'tcompletelystochasticandiscompletelyrelatedtothestockprice,soitisnotidealtoovercomethe\volatilitysmilee®ect.Thus,Chapter6extendstherisky-assetpricestoHestonstochasticvolatilitymodel,studiestheoptimalinvestmentproblemofthede¯ned-contributionpensionundermaximizingtheexpectedpowerutility,andobtainstheexplicitsolutionbytheHJBequation,powertransformationandvariablereplacementtechniques.KEYWORDS:pensionfunds;expectedutility;stochasticsalary;a±neinterestratesmodel;constantelasticityofvariance(CEV)model;Leg-endretransformation;Hamilton-Jacobi-Bellman(HJB)equation;optimalin-vestmentv...............................................11.1.........................................11.2..............................................21.2.1.......................21.2.2.........