Stochastic differential equations harvesting optimization with stochastic prices: Formulation and numerical solution

Abstract

This work aims to achieve optimal harvesting in a random setting with a stochastic price structure. We use a general growth function to model the harvested population, a geometric Brownian motion to model price change, and add fluctuations in the interest rate over time to complete the analysis. Following this, we make use of the stochastic dynamic programming technique in order to obtain the Hamilton–Jacobi–Bellman equation, which ultimately results in the optimal combination of profit and effort. We employ the Crank–Nicolson discretization approach to obtain a numerical solution to the Hamilton–Jacobi–Bellman partial differential equation. For application purposes, we consider a Gompertz growth model and realistic data based on the Bangladesh shrimp.