Ergodic singular stochastic control motivated by the optimal sustainable exploitation of an ecosystem

We derive the explicit solution to a singular stochastic control problem of the monotone follower type with an expected ergodic criterion as well as to its counterpart with 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 and the running payoff function, we show that both performance criteria give rise to the same optimal long-term average rate as well as to the same optimal strategy, which is of a threshold type. We solve the two problems by first constructing a suitable solution to their associated Hamilton-Jacobi-Bellman (HJB) equation, which takes the form of a quasi-variational inequality with a gradient constraint.