Before revision:  Assuming the diffusion component of the process is non-degenerate and a mild assumption on the singularity of the Levy measure, this paper shows that the value function is smooth in the continuation region for problems with either finite or infinite variation jumps .
After revision:  Assuming the diffusion component of the process is nondegenerate and a mild assumption on the singularity of the Levy measure, this paper shows that the value function is smooth in the continuation region for problems with either finite or infinite variation jumps .