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This paper investigates a singular stochastic control problem for amulti-dimensional regime-switching diffusion process confined in an unboundeddomain. The objective is to maximize the total expected discounted rewards fromexerting the singular control. Such a formulation stems from application areassuch as optimal harvesting multiple species and optimal dividends paymentsschemes in random environments. With the aid of weak dynamic programmingprinciple and an exponential transformation, we characterize the value functionto be the unique constrained viscosity solution of a certain system of couplednonlinear quasi-variational inequalities. Several examples are analyzed indetails to demonstrate the main results.

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