A Solution Technique for Lévy Driven Long Term Average Impulse Control Problems

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.