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

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, 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. Key words. constrained viscosity solution, regime-switching diffusion, singular stochastic control, weak dynamic programming principle, quasi-variational inequality