Optimal exploitation of a resource with stochastic population dynamics and delayed renewal

In this work, we study an optimization problem arising when managing a renewable resource in finite time. The resource is assumed to evolve according to a logistic stochastic differential equation. The manager harvests partially the resource at any time and sell it at a stochastic market price. She equally decides to renew part of the resource but uniquely at deterministic times. However, we realistically assume that there is a delay in the renewing order. By using the dynamic programming theory, we characterize our value function as the unique solution to a PDE. To complete our study, we give an algorithm to compute the value function and optimal strategy. Some numerical illustrations are also provided.