Solutions of BSDE's with jumps and quadratic/locally Lipschitz generator

Inspired by utility optimization problems in finance, in this paper we prove the existence of the solution of a class of BSDE's driven by a Brownian motion and a jump process, whose generator shows quadratic growth in the Brownian component and exponential growth with respect to the jump term. Existence and uniqueness of the solution is established first for bounded terminal value, then we extend the existence result to the unbounded case, under appropriate hypotheses.