带有gamma限制的倒向随机微分方程及在最优复制中的应用new

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1、http://www.paper.edu.cnBackwardStochasticDifferentialEquationunderGammaConstraintandApplicationstoOptimalReplicationYuhongXu∗(DepartmentofMathematics,ChinaUniversityofMiningandTechnologyXuzhou,221008,P.R.China)AbstractThispaperprovesanexistencetheoremforaclassofbackwa

2、rdstochasticdifferentialequationwithdoubleintegralofItˆotype,whichleadstoadoublestochasticintegralrepresentationforsquareintegrablefunctionalsofBrownianmotion;wethengiveanapplicationtooptimalreplicationproblemwithgammaconstraints,itisshownthatthesocalled“gamma”isinfac

3、tthesecondorderMalliavinderivativeofthewealthprocessandgammaconstrainthereisprovedtobetheformalone;moreover,someresultsonlinearchaosexpansionandfullynonlinearmathematicalexpectationareobtained.keyword:Backwardstochasticdifferentialequation;doublestochasticintegrationr

4、epresen-tation;optimalreplicationundergammaconstraints;chaosexpansion1IntroductionBackwardstochasticdifferentialequation(BSDE)undergammaconstraintoriginallycomesfromtheproblemofsuper-replicationundergammaconstraint.Inviewofpractice,fewderivativesaretradedcontinuously.

5、ForthehedgingportfolioZ,alarge(dZt)meansalargerisktothetraders.Sowewanttocontrol“gamma”whichdenotesthevariationofthehedgingportfoliowithinaboundedinterval.Usually,thisconstraintongammaleadstonosolutionsforthecorrespondingBlack-scholesmodel.Super-replicationisanaltern

6、ative.Theproblemofsuper-replicationundergammaconstraintwasfirsttreatedinBroadie,Cvitani´candSoner(1998)andlaterformulatedpreciselyinSonerandTouzi(2000)withuppergammabound.ThegeneralcaseisreferredtoCheridito,SonerandTouzi(2005a,2005b)andSonerandTouzi(2007).Super-replic

7、ationundergammaconstraintinabovepaperareapproachedbydynamicprogrammingprinciples.ThetoolweuseinthispaperisthewellknownBSDEariseinstochasticcontrolandmathematicalfinance.BSDEisaconvenienceandpowerfultooltodealwithhedgingproblemsunderconstraints.Theinitialmotivationfort

8、hispaperistostudytheexistenceofsolutiontosecondorderback-wardstochasticdifferentialequation(2BSDE)whichisaclassofForward-BackwardSDE