A Solution Technique for L\'evy Driven Long Term Average Impulse Control Problems

TL;DR

This paper develops a method for solving long-term average impulse control problems driven by Lévy processes, characterizing the value via stopping problems and providing explicit strategies under general conditions.

Contribution

It introduces a step-by-step solution technique for impulse control problems with Lévy processes, including explicit threshold determination and strategy construction.

Findings

Characterization of control problem as a stopping problem

Explicit thresholds given by roots of an auxiliary function

Demonstration through various example solutions

Abstract

This article treats long term average impulse control problems with running costs in the case that the underlying process is a L\'evy process. Under quite general conditions we characterize the value of the control problem as the value of a stopping problem and construct an optimal strategy of the control problem out of an optimizer of the stopping problem if the latter exists. Assuming a maximum representation for the payoff function, we give easy to verify conditions for the control problem to have an $\left(s,S\right)$ strategy as an optimizer. The occurring thresholds are given by the roots of an explicit auxiliary function. This leads to a step by step solution technique whose utility we demonstrate by solving a variety of examples of impulse control problems.