Viscosity Solution of Optimal Stopping Problem for Stochastic Systems with Bounded Memory

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1、ViscositySolutionofOptimalStoppingProblemforStochasticSystemswithBoundedMemory∗Mou-HsiungChang†TaoPang‡MoustaphaPemy§April5,2012AbstractWeconsiderafinitetimehorizonoptimalstoppingproblemforasys-temofstochasticfunctionaldifferentialequationswithaboundedmemory.Undersomesufficientlysmoothconditions,aHamilt

2、on-Jacobi-Bellman(HJB)variationalinequalityforthevaluefunctionisderivedviadynam-icalprogrammingprinciple.ItisshownthatthevaluefunctionistheuniqueviscositysolutionoftheHJBvariationalinequality.KEYWORDS:optimalstopping,stochasticcontrol,stochasticfunctionaldif-ferentialequations.AMS2000subjectclassific

3、ations:primary60G40,60G20;secondary60H30,93E20∗TheresearchofthispaperispartiallysupportedbyagrantW911NF-04-D-0003fromtheU.S.ArmyResearchOffice†MathematicalSciencesDivision,U.S.ArmyResearchOffice,P.O.Box12211,RTP,NC27709,USA.Email:mouhsiung.chang@us.army.mil‡DepartmentofMathematics,NorthCarolinaStateUniv

4、ersity,Raleigh,NC27695,USA.Email:tpang@ncsu.edu(CorrespondingAuthor)§DepartmentofMathematics,TowsonUniversity,Towson,MD21252-0001,USA.Email:mpemy@towson.edu11IntroductionOptimalcontrolandoptimalstoppingproblemsoverafiniteoraninfinitetimehorizonforItˆo’sdiffusionprocessesariseinmanyareasofscience,engine

5、ering,andfinance(seee.g.FlemingandSoner[14],Øksendal[35],Shiryaev[39],KarazasandShreve[19]andreferencescontainedtherein).ThevaluefunctionoftheseproblemsarenormallyexpressedasaviscosityorageneralizedsolutionofaHamilton-Jacobi-Bellmanequation(HJBE)oraHamilton-Jacobi-Bellmanvariationalinequality(HJBVI)t

6、hatinvolvesasecondorderparabolicorellipticpartialdifferentialequationinafinitedimensionalEuclideanspace(seee.g.Lions[30]and[32]).Inanattempttoachievebetteraccuracyandtoaccountforthedelayedeffectofthestatevariablesinthemodellingofrealworldstochasticcontrolproblems,thestochasticdelayequationsandcontrolle

7、dstochasticdelayequationshavebeenthesubjectofintensivestudiesinrecentyearsbymanyresearcherssuchasElsanousi[11],Larrssen[25],Elsanousietal[13],ØksendalandSulem[37],LarrssenandRisebro[27],andBauerandRie