Singular control with state-dependent costs for Lévy processes

We study a discounted singular stochastic control problem driven by a general Lévy 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.