Impulse control and expected suprema

TL;DR

This paper studies impulse control problems for Markov processes, providing explicit solutions and characterizing optimal thresholds through a representation of excessive functions using expected suprema.

Contribution

It introduces a method to explicitly solve impulse control problems for general Markov processes using a novel representation of excessive functions.

Findings

Optimal impulse times are of threshold type.

Optimal thresholds are characterized by nonlinear equations.

Explicit solutions are derived for a broad class of processes.

Abstract

We consider a class of impulse control problems for general underlying strong Markov processes on the real line, which allows for an explicit solution. The optimal impulse times are shown to be of threshold type and the optimal threshold is characterized as a solution of a (typically nonlinear) equation. The main ingredient we use is a representation result for excessive functions in terms of expected suprema.