Lp solution of backward stochastic differential equations driven by a marked point process

We obtain existence and uniqueness in Lp, p>1 of the solutions of a backward stochastic differential equation (BSDE for short) driven by a marked point process, on a bounded interval. We show that the solution of the BSDE can be approximated by a finite system of deterministic differential equations. As application, we address an optimal control problem for point processes of general non-Markovian type and show that BSDEs can be used to prove existence of an optimal control and to represent the value function.