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 . Moreover , the smooth-fit property is shown via the global regularity of the value function .
After 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 of problems on an unbounded domain with infinite activity jumps is W^{2,1 , the smooth-fit property is shown via the global regularity of the value function .