Regularity of the Optimal Stopping Problem for Jump Diffusions

TL;DR

This paper proves that the value function for an optimal stopping problem involving jump diffusions is sufficiently smooth under certain conditions, ensuring the smooth-fit property holds, which advances understanding of such stochastic control problems.

Contribution

It establishes the regularity of the value function for jump diffusion optimal stopping problems under mild assumptions, extending previous results to more general jump processes.

Findings

Value function belongs to $W^{2,1}_{p, loc}$ space.

Smooth-fit property holds for the problem.

Results apply to jump diffusions with finite or infinite variation jumps.

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