An optimal multibarrier strategy for a singular stochastic control problem with a state-dependent reward

TL;DR

This paper investigates optimal multibarrier strategies for a stochastic control problem with state-dependent rewards, providing conditions for optimality in spectrally negative Lévy processes and Brownian motion cases, along with a computational algorithm.

Contribution

It establishes sufficient conditions for optimality of one- and multi-barrier strategies in specific stochastic processes, extending previous results and offering a practical computation method.

Findings

Optimal one-barrier strategy for spectrally negative Lévy processes.

Optimal (2n+1)-barrier strategy for Brownian motion with drift.

Algorithm for computing barrier levels in the Brownian motion case.

Abstract

We consider a singular control problem that aims to maximize the expected cumulative rewards, where the instantaneous returns depend on the state of a controlled process. The contributions of this paper are twofold. Firstly, to establish sufficient conditions for determining the optimality of the one-barrier strategy when the uncontrolled process $X$ follows a spectrally negative L\'evy process with a L\'evy measure defined by a completely monotone density. Secondly, to verify the optimality of the $(2n+1)$-barrier strategy when $X$ is a Brownian motion with a drift. Additionally, we provide an algorithm to compute the barrier values in the latter case.