On solutions of backward stochastic differential equations with jumps and applications

For backward stochastic differential equation (BSDE) with jumps and with non-lipschitzian coefficient the existence and uniqueness of an adapted solution is obtained. By generalizing the existence result on partial differential and integral equations (PDIE) and Ito formula to the functions with only first and second Sobolev derivatives the probabilistic interpretation for solutions of PDIE (a new Feynman-Kac formula) by means of solutions of BSDE with jumps is got. With the help of this formula a new existence and uniqueness result for the solution of PDIE with non-lipschitzian force is obtained. The convergence theorems of solutions to BSDE and PDIE are also derived.