The value function of an optimal stopping problem for a process with L\'{e}vy jumps is known to be a generalized solution of a variational inequality. 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 obstacle problems on an unbounded domain with infinite activity jumps is W^{2,1}_{p, loc}  .
The value function of an optimal stopping problem for a process with L\'{e}vy jumps is known to be a generalized solution of a variational inequality. 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 obstacle problems on an unbounded domain with finite/infinite variation jumps is in W^{2,1}_{p, loc} . As a consequence, the smooth-fit property holds .
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infinite activity jumps is
finite/infinite variation jumps is in
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. As a consequence, the smooth-fit property holds