On solutions of backward stochastic differential equations with jumps and stochastic control

We relax conditions on coefficients given in [7] for the existence of solutions to backward stochastic differential equations (BSDE) with jumps. Counter examples are given to show that such conditions can not be weakened further in some sense. The existence of a solution for some continuous BSDE with coefficients b(t, y, q) having a quadratic growth in q, having a greater than linear growth in y, and are unbounded in y belonging to a finite interval, is also obtained. Then we obtain an existence and uniqueness result for the Sobolev solution to some integro-differential equation (IDE) under weaker conditions. Some Markov properties for solutions to BSDEs associated with some forward SDEs are also discussed and a Feynman-Kac formula is also obtained. Finally, we obtain probably the first results on the existence of non-Lipschitzian optimal controls for some special stochastic control problems with respect to such BSDE systems with jumps, where some optimal control problem is also explained in the financial market.