On singular control for L\'evy processes

TL;DR

This paper establishes the optimality of barrier strategies in singular control problems driven by a broad class of Lévy processes, extending previous results limited to Brownian motion or Lévy processes with one-sided jumps.

Contribution

It proves the optimality of barrier strategies for convex running costs under general Lévy processes, broadening the scope of classical singular control results.

Findings

Barrier strategies are optimal for a wide class of Lévy processes.

Convexity of the running cost function is key to the optimality.

Results extend previous work limited to Brownian and one-sided jump Lévy processes.

Abstract

We revisit the classical singular control problem of minimizing running and controlling costs. The problem arises in inventory control, as well as in healthcare management and mathematical finance. Existing studies have shown the optimality of a barrier strategy when driven by the Brownian motion or L\'evy processes with one-side jumps. Under the assumption that the running cost function is convex, we show the optimality of a barrier strategy for a general class of L\'evy processes.