Backward Stochastic Differential Equations And Malliavin Derivatives

This paper mainly refers to the dissertation "Backward Stochastic Differential E-quations in Finance" done by Shi[1].In this dissertation,we briefly introduce the basic knowledge and applications of the backward stochastic differential equations(BSDE).Liang[2]demonstrated that BSDE may be reformulated as ordinary functional equations on certain path spaces.Shi applied the skill to study the class of BSDE dYtj =—fj(t,Yt,L(M)t)dt + dMtj,YTj ??j,where L is a non-linear mapping that sends M to an adapted process L(M),and correc-tion term M is a matringale to be determined.Under certain technical conditions,the existence and uniqueness of L2 solutions were proved in Liang's paper.Further more,it has been successfully generalized into Lp solutions under these conditions by Shi.What'a more,comparison theorem of the above BSDE is established.Then,the Malliavin derivatives of L2 solutions of following BSDE is introduced dYt=—f(t,Yt,Zt)dt + Zt*dBt,YT=?j.Based on our theorems,some famous theorems in other literature were revisited and proved.Last but not least,some applications in finance are introduced to explain the di-rection of our theorems.And the Malliavin derivative of L2 solutions together with the comparison theorem is applied to the European option pricing.