Before revision:  Assuming the diffusion component of the process is nondegenerate and a mild assumption on the singularity of the L\'{e}vy measure, this paper shows that the value function of  problems on an unbounded domain with infinite activity jumps is W^{2,1}_{p, loc} . As a result, the smooth-fit property holds and the value function is C^{2,1 .
After revision:  Assuming the diffusion component of the process is nondegenerate and a mild assumption on the singularity of the L\'{e}vy measure, this paper shows that the value function of problems on an unbounded domain with infinite activity jumps is W^{2,1}_{p, loc} .