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1、AStochasticPortfolioOptimizationModelwithBoundedMemoryMou-HsiungChangyTaoPangzYipengYangxMarch25,2010AbstractThispaperconsidersaportfoliomanagementproblemofMerton'stypeinwhichtheriskyassetreturnisrelatedtothereturnhistory.Theproblemismodeledbyastochasticsystemwithdelay.Theinvestor'sg
2、oalistochoosetheinvestmentcontrolaswellastheconsumptioncontroltomaximizehistotalexpected,discountedutility.Undercertainsituations,wederivetheexplicitsolutionsina nitedimensionalspace.Keywords:Stochasticdelayequations,optimalstochasticcontrol,HJBequationAMS2000subjectclassi cations:Pri
3、mary:34K50;Secondary:93E20,49L201Introduction.Inthispaper,weconsiderastochasticportfoliomanagementprobleminwhichthehistory(delay)oftheportfolioperformanceistakenintoconsideration.IntheclassicalMerton'smodel,thepriceprocessoftheriskyassetismodeledbyageometricBrownianmotion,whichisaMark
4、ovianprocess.Inotherwords,theinvestorwillmakehisinvestmentdecisionsonlybasedonthecurrentinformation(riskyassetprice,portfoliovalue,etc.)andwillnotconsiderthehistoryofthestockpricesorportfoliovalues.However,intherealworld,investorstendtolookatthehistoricperformanceoftheriskyassetorthei
5、rportfoliosbeforetheymaketheinvestmentdecision.Forexample,ifthepricehasbeenincreasingalotrecently,moreinvestorstendtoinvestmoremoneyontothisasset,whichwillpushthepriceevenhigher.Ontheotherhand,ifthepricehasbeendecreasingalot,moreinvestorstendtoselltheassetandinvestonotherassets,whichw
6、illdrivethepricetogodownfurther.Tomodelthisphenomenon,weuseastochasticdelayedequationtodescribethevaluechangeoftheinvestor'sportfolio,whichincludesariskyassetandarisklessasset.Inparticular,weconsideraportfoliowhichincludesariskyassetandarisklessasset.Thevalueoftheportfoliofollowsastoc
7、hasticprocessX(t)(see(10))whichdependsonthefollowingdelayTheresearchofthispaperispartiallysupportedbyagrantW911NF-04-D-0003fromtheU.S.ArmyResearchOceyMathematicsDivision,U.S.ArmyResearchOce,P.O.Box12211,RTP,NC27709,USA,mouhsi-ung.chang@us.army.milzDepartmentofMathematicsandCenterfo
8、rRese
