Impulse Control of Multi-dimensional Jump Diffusions in Finite Time Horizon

TL;DR

This paper studies impulse control problems for multi-dimensional jump diffusions over finite time, establishing the dynamic programming principle, viscosity solutions for the HJB equation, and regularity of the value function under specific jump and diffusion conditions.

Contribution

It rigorously formulates the DPP, proves the value function as a unique viscosity solution with regularity, and extends the theory to jump diffusions with infinite activity and finite variation.

Findings

Established the Dynamic Programming Principle for the problem.

Proved the value function is a viscosity solution to the HJB equation.

Showed the value function has $W_{loc}^{(2,1),p}$ regularity under certain conditions.

Abstract

This paper analyzes a class of impulse control problems for multi-dimensional jump diffusions in the finite time horizon. Following the basic mathematical setup from Stroock and Varadhan \cite{StroockVaradhan06}, this paper first establishes rigorously an appropriate form of Dynamic Programming Principle (DPP). It then shows that the value function is a viscosity solution for the associated Hamilton-Jacobi-Belleman (HJB) equation involving integro-differential operators. Finally, under additional assumptions that the jumps are of infinite activity but are of finite variation and that the diffusion is uniformly elliptic, it proves that the value function is the unique viscosity solution and has $W_{loc}^{(2,1),p}$ regularity for $1< p< \infty$