Regularity of the optimal stopping problem for jump diffusions

The value function of an optimal stopping problem for jump diffusions is known to be a generalized solution of a variational inequality. Assuming that the diffusion component of the process is nondegenerate and a mild assumption on the singularity of the L´evy measure, this paper shows that the value function of this optimal stopping problem on an unbounded domain with finite/infinite variation jumps is in W2;1 p;loc with p 2 (1;1). As a consequence, the smooth-fit property holds