Regularity of the Optimal Stopping Problem for Jump Diffusions

Abstract

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 Lévy measure, this paper shows that the value function of this optimal stopping problem on an unbounded domain with finite/infinite variation jumps is in $W_{p,loc}^{2,1}$ with $p\in (1,\infty )$. As a consequence, the smooth-fit property holds.

To Appear in the SIAM Journal on Control and Optimization