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

TL;DR

This paper models the optimal management of a renewable resource with stochastic growth and delayed renewal, providing a PDE-based framework, an algorithm for strategy computation, and numerical illustrations.

Contribution

It introduces a novel stochastic control model incorporating delayed renewal decisions and develops a PDE characterization and computational algorithm for optimal strategies.

Findings

Derived PDE for the value function of the resource management problem.

Developed an algorithm to compute optimal harvesting and renewal strategies.

Provided numerical examples demonstrating the effectiveness of the approach.

Abstract

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.