On Singular Stochastic Control Problems for Diffusions with Jumps

We consider some cases of control problems for diffusion processes with jumps when the payoff functional does not depend explicitly on the control. We prove the continuity of the optimal cost functions and we give a characterization of this cost with a quasi-variational inequality interpreting the problem as the limit of an impulse control problem when the cost of impulse tends to zero. Moreover, we show the existence of an optimal control for some particular situations.