Coupled Forward-Backward Stochastic Differential Equations Driven By The Continuous Martingale

The linear backward stochastic differential equation was first proposed by Bismut in 1978.Then Pardoux and Peng first solved the existence and uniqueness theorem of the solution of the nonlinear BSDE under Lipschitz condition in 1990.From then on,many people make further study on BSDE and its applications in mathematical finance, stochastic control,partial differential equation(PDE),stochastic differential games and economy,which develop BSDE further.Now the theory of BSDE is not only widely considered as the main tool of study financial mathematial(for example,the problem of pricing of options and derivative securities) but also the efficient tool of studying stochastic control,stochastic games,the problem of probabilistic represention of solution of nonlinear PDE and so on.The classical BSDE is driven by Brownian Motion, but Brownian Motion is an ideal stochastic model which ristricts the applications of the classical BSDE.In this paper,we generalize the classical BSDE essentially,its noise source by the Brownian Motion for continuous martingale,discussed some issues of Coupled Forward-Backward Stochastic Differential Equations driven by the continuous martingale.Coupled Forward-Backward Stochastic Differential Equations with Brownian Motion was first proposed by Antonelli.Since S.Peng and Zhen.Wu proposed the existence and uniqueness theorem of the solution of Coupled Forward-Backward Stochastic Differential Equations,many scholars have made the unrenmitting effort in this aspect.This paper based on the predecessor's work,first gives the condition of the existence and uniqueness theorem of the solution of Coupled Forward-Backward Stochastic Differential Equations driven by tne continuous martingale,and proves it by using purely probabilistic method.Considering these questions are still faced with many challenges,such as,general martingale is not Martingale said Theorem nature,in other words,there can be no assure BSDE the existence of solutions.On this issue,we will assume that Martingale has said can be expected to be its nature.Feynman-kac formula is a important formula of Stochastic Analysis.Peng provides largest category of a second-order linear parabolic partial differential system with a probabilistic interpretation by using classic BSDE.The results promote famous Feynman-kac formula extended to the nonlinear case,Nonlinear Feynman-kac that formula,It is a common BSDE and the solution of nonlinear partial differential equations of the correspondence between solutions.In this paper,using the special nature of Ocone martingale,discussed by Coupled Forward-Backward Stochastic Differential Equations driven by the continuous martingale and contact it with a kind of one-algebra The quasi-linear equations of parabolic partial differential equations systems to link up also given a formula Feynman-kac promotion.