On Singular Control Problems with State Constraints and Regime-Switching: A Viscosity Solution Approach

TL;DR

This paper studies a complex stochastic control problem involving regime-switching diffusions with state constraints, characterizing the value function as a unique viscosity solution to coupled inequalities, with applications in ecology and finance.

Contribution

It introduces a novel viscosity solution framework for singular control problems with regime-switching and state constraints, extending existing theories.

Findings

Value function characterized as unique viscosity solution

Application to multi-species harvesting and dividend schemes

Detailed examples demonstrating main results

Abstract

This paper investigates a singular stochastic control problem for a multi-dimensional regime-switching diffusion process confined in an unbounded domain. The objective is to maximize the total expected discounted rewards from exerting the singular control. Such a formulation stems from application areas such as optimal harvesting multiple species and optimal dividends payments schemes in random environments. With the aid of weak dynamic programming principle and an exponential transformation, we characterize the value function to be the unique constrained viscosity solution of a certain system of coupled nonlinear quasi-variational inequalities. Several examples are analyzed in details to demonstrate the main results.