On a Class of Singular Stochastic Control Problems for Reflected Diffusions

TL;DR

This paper studies optimal control strategies for reflected diffusions with state-dependent rewards and reflection costs, revealing different optimal behaviors based on reward properties using stochastic control and diffusion theory.

Contribution

It introduces a framework for singular control of reflected diffusions with state-dependent rewards and costs, analyzing how reward properties influence optimal strategies.

Findings

Different types of optimal strategies depending on reward functions

Application of stochastic control techniques to reflected diffusions

Insights into cost-reward trade-offs in reflected stochastic processes

Abstract

Reflected diffusions naturally arise in many problems from applications ranging from economics and mathematical biology to queueing theory. In this paper we consider a class of infinite time-horizon singular stochastic control problems for a general one-dimensional diffusion that is reflected at zero. We assume that exerting control leads to a state-dependent instantaneous reward, whereas reflecting the diffusion at zero gives rise to a proportional cost with constant marginal value. The aim is to maximize the total expected reward, minus the total expected cost of reflection. We show that depending on the properties of the state-dependent instantaneous reward we can have qualitatively different kinds of optimal strategies. The techniques employed are those of stochastic control and of the theory of linear diffusions.