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 Lévy 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 (s, S) 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.

General Optimal Stopping with Linear Costs

“…First we give a brief overview on literature regarding maximum representations for general Markov processes, thereafter we focus on explicit obtainability in the case that the underlying process is a Lévy process. This construction has been developed in [CS19] in a similar setting, therefore we will leave out the proofs if they are already given in said work.…”

“…After that we first discuss existence of such a function and also in the case of Lévy processes give a construction (analogous to the discrete time case with help of the ladder height process). We follow the line of argument used in [CS19] for a related problem arising in impulse control.…”

“…First, we will give sufficient conditions for an f as in (2) to exist and thereafter take some steps to the (semi-)explicit obtainability in interesting cases. This theory was developed in [CS19] and the detailed proofs can be found therein.…”

“…[CR61], and used extensively throughout the following decades. More recently, the power of this easy line of argument for more advanced problems was rediscovered, see [Chr17,CI19], or [CS19] in the context of impulse control problems. One idea in these work was to embed monotone problems in a priori non-monotone stopping settings.…”