Existence And Uniqueness For A Backward Stochastic Differential Equation Whose Generator Is Montel-Tonelli

Backward stochastic differenial equation(BSDE)was given the general for-m by Pardoux-Peng in1990.In addition,they have proved the existence and uniqueness,the BSDE theory just started its booming development.The BSDE is the following forms: Yt=ξ+∫tTg(s,Ys,Zs)ds-∫tTZsdBs,t∈[O,T] where ξ represents the terminal random variable,T>0is a fixed terminal time,(Bt)t∈[O,T] represents the d-dimensional Brownian motion and g is called the generator of the BSDE.The solution of BSDE is a pair of adapted processes (Y.Z.)that makes the above equality hold.Pardoux-Peng assumed that g is Lipschitz continuous about y,z,and since then,lots of papers were devoted to the relaxation of the conditions on g.For instance, J.P.Lepeltier,Jia assumed that g is uniformly continuous in z,uniformly with respect to (ω,t)and indepen-dent of y,the solution to the backward stochastic differential equation(BSDE)with generator g,is unique.In this paper,we are interested in solving one-dimensional backward s-tochastic differential equation(BSDE) whose generator satisfy the Montel-Tonelli condition. We establish an existence and uniqueness result of solution for this kind of BSDE.The significance of existence and uniqueness is that under the condition of a given terminal value for the particular form of stochastic differ-ential equation,we can prove that there exist a pair of adapt solution,and the adapt solution is unique.Thus we can use the determine methods,strategies to solve the problem of uncertainty,or deal with the problem of random uncer-tainty to optimal as a result.The existence and uniqueness of solution implies that if the target at time T is given,we can calculate what the starting point has to make the system to achieve the desired goal.It shows the importance of the existence and uniqueness of backward stochastic differential equation-s.Backward stochastic differential equation(BSDE)has been widely applied in stochastic control,in Partial differential Equations,in Mathematical Finance,in Economy and so on.The first part of this article,introduces some basic concepts.In the second part of this article,introduce some of the major report of the existence and uniqueness of backward stochastic differential equation.In the third part of this article, we show that the main conclusions.