Optimal harvesting from a population in a stochastic crowded environment

We study the (Ito) stochastic differential equation dX(t) = rX(t)(K - X(t))dt + aX(t)(K - X(t))dB(t), X₀ = x > 0 as a model for population growth in a stochastic environment with finite carrying capacity K > 0. Here r and a are constants and B(t) denotes Brownian motion. If r ≤ 0, we show that this equation has a unique strong global solution for all x > 0 and we study some of its properties. Then we consider the following problem: What harvesting strategy maximizes the expected total discounted amount harvested (integrated over all future times)? We formulate this as a stochastic control problem. Then we show that there exists a constant optimal 'harvest trigger value' x* ε (0, K) such that the optimal strategy is to do nothing if X(t) < x* and to harvest X(t) - x* if X(t) > x*. This leads to an optimal population process X(t) being reflected downward at x*. We find x* explicitly.