MIXED OPTIMAL STOPPING AND STOCHASTIC CONTROL PROBLEMS WITH SEMICONTINUOUS FINAL REWARD FOR DIFFUSION PROCESSES

We consider mixed control problems for diffusion processes, i.e. problems which involve both optimal control and stopping. The running reward is assumed to be smooth, but the stopping reward need only be semicontinuous. We show that, under suitable conditions, the value function w has the same regularity as the final reward g, i.e. w is lower or upper semicontinuous if g is. Furthermore, when g is l.s.c., we prove that the value function is a viscosity solution of the associated variational inequality.