On Solutions of Backward Stochastic Differential Equations with Jumps and Stochastic Control
We relax conditions on coefficients given in  for the existence of solu-tions 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.
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