Singular stochastic control problems motivated by the optimal sustainable exploitation of an ecosystem

We derive the explicit solutions to singular stochastic control problems of the monotone follower type with (a) an expected discounted criterion, (b) an expected ergodic criterion, and (c) a pathwise ergodic criterion. These problems have been motivated by the optimal sustainable exploitation of an ecosystem, such as a natural fishery. Under general assumptions on the diffusion coefficients, the discounting rate function, the running payoff function, and the marginal profit of control action, we show that the optimal strategies are of a threshold type. We solve the three problems by first constructing suitable solutions to their associated HJB equations, which take the form of quasi-variational inequalities with gradient constraints. In the cases of the ergodic control problems, we also use a suitable new variational argument. Furthermore, we establish the convergence of the solution of the discounted control problem to the solution of the ergodic control problems as the discounting rate function tends to 0 in an Abelian sense.