  This work deals with backward stochastic differential equation (BSDE) with
random marked jumps, and their applications to default risk. We show that these
BSDEs are linked with Brownian BSDEs through the decomposition of processes
with respect to the progressive enlargement of filtrations. We show that the
equations have solutions if the associated Brownian BSDEs have solutions. We
also provide a uniqueness theorem for BSDEs with jumps by giving a comparison
theorem based on the comparison for Brownian BSDEs. We give in particular some
results for quadratic BDSEs. As applications, we study the pricing and the
hedging of a European option in a complete market with a single jump, and the
utility maximization problem in an incomplete market with a finite number of
jumps.
