The Existence Of Solution For RBSDE With L～2 -Obstacles And Continuous Coefficient

In 1990, Pardoux and Peng introduced the following backward stochastic differential equation (BSDE):and proved there exists unique adopted solution to this equation. From then on, lots of scholars have interested in this field and now this result has been applied to finance, economic and other branches of mathematics.Now this result has already been developed. El Karoui, Kapoudjian, Pardoux, Peng, Quenez [1997a] considered for the first time that BSDE with obstacle, that is to say the solution of BSDE Y is bigger than a given process. And they proved that if the coefficient is Lipschitz and the obstacle is continuous, then the solution exists and is unique.In this paper, we consider RBSDE with continuous coefficient and L_F~2 obstacle. For BSDE with one lower obstacle, the solution is a triple (Y, Z, A), where A is increasing, Y_t≥L_t a.e. a.s. satisfying the following BSDE:and the generalized Skorohod condition:for each L*∈S_F~2 such that L_t≤L*_t≤Y_t, a.e. a.s. Lepeltier and San Martin [1997] showed that for a continuous function f, there exists a sequence of Lipschitz function f_m that converges to f, as m→∞. With this result we can treat with the continuous coefficient. For the obstacle, we follow the penalization method, with the monotonic theorem of BSDE we then have the existence of solution for RBSDE. Moreover, although we don't have the uniqueness result, we can prove that the solution got from our method is the smallest one, and then we have the comparison theorem of RBSDE.At last, we consider RBSDE with two obstacles. That is to say the solution of BSDE is between two given process: L≤Y≤U. The solution of RBSDE is a quadruple (Y, Z, A, K), satisfying the following BSDE:where A, K is increasing, and the solution satisfies the Skorohod condition:For any L*, U*∈S_F~2 such that L_t≤L*_t≤Y_t≤U*_t≤U_t, a.e. a.s.