Optimal Exploitation of a Resource with Stochastic Population Dynamics and Delayed Renewal

In this work, we study the optimization problem of a renewable resource in finite time. The resource is assumed to evolve according to a logistic stochastic differential equation. The manager may harvest partially the resource at any time and sell it at a stochastic market price. She may equally decide 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 may obtain the PDE characterization of our value function. To complete our study, we give an algorithm to compute the value function and optimal strategy. Some numerical illustrations will be equally provided.