Singular control with state-dependent costs for L\'evy processes

TL;DR

This paper analyzes a stochastic control problem driven by Le9vy processes, establishing conditions for optimal reflecting barrier strategies and linking them to optimal stopping problems for explicit solutions.

Contribution

It introduces a novel connection between singular control with state-dependent costs and optimal stopping, enabling explicit solutions in complex Le9vy-driven models.

Findings

Optimal reflecting barrier strategies are characterized via a verification theorem.

The optimal control threshold coincides with the optimal stopping boundary.

Explicit solutions are obtained in several practical cases, including pollution abatement.

Abstract

We study a discounted singular stochastic control problem driven by a general L\'evy process, where the objective is to minimize a cost functional composed of a running cost and a control cost that depends on the current state of the process. We first establish a Hamilton Jacobi Bellman (HJB)-type verification theorem providing sufficient conditions under which a reflecting barrier strategy is optimal and characterizing the value function. Our main contribution is to connect this control problem with an associated optimal stopping problem: we prove that the optimal reflection threshold coincides with the optimal stopping boundary of the auxiliary problem. This connection allows us to characterize the optimal strategy through probabilistic tools and leads to explicit or semi-explicit solutions in several relevant cases. We illustrate the results with several examples, including an application to pollution abatement.