Optimality of doubly reflected Levy processes in singular control

TL;DR

This paper analyzes a class of two-sided singular control problems involving spectrally negative Levy processes, establishing conditions under which a double barrier strategy is optimal and expressing solutions via scale functions.

Contribution

It provides a sufficient condition for the optimality of double barrier strategies in singular control of Levy processes, especially when the running cost is convex.

Findings

Double barrier strategy is optimal under certain conditions.

Optimal value function expressed using scale functions.

Numerical examples confirm analytical results.

Abstract

We consider a class of two-sided singular control problems. A controller either increases or decreases a given spectrally negative Levy process so as to minimize the total costs comprising of the running and control costs where the latter is proportional to the size of control. We provide a sufficient condition for the optimality of a double barrier strategy, and in particular show that it holds when the running cost function is convex. Using the fluctuation theory of doubly reflected Levy processes, we express concisely the optimal strategy as well as the value function using the scale function. Numerical examples are provided to confirm the analytical results.