On the Impulse Control of Jump Diffusions

TL;DR

This paper extends the analysis of impulse control problems for jump diffusions to include cases with infinite activity and variation, proving regularity of the value function in a Sobolev space.

Contribution

It generalizes previous work by establishing regularity results for impulse control problems with more complex jump processes.

Findings

Value function belongs to W^{2,p}_{loc}(R^n)

Extends regularity results to infinite activity and variation jumps

Builds on prior work by Davis, Guo, and Wu (2010)

Abstract

Regularity of the impulse control problem for a non-degenerate $n$-dimensional jump diffusion with infinite activity and finite variation jumps was recently examined by Davis, Guo, and Wu (SICON 2010). Here we extend the analysis to include infinite activity and infinite variation jumps. More specifically, we show that the value function $u$ of the impulse control problem satisfies $u\in W^{2,p}_{\mathrm{loc}}(\mathbb{R}^{n})$.