Optimal harvest under a Gilpin–Ayala model driven by the Hawkes process

Abstract

This paper analyzes the optimal effort for a risk-averse fisherman where the biomass process follows a Hawkes jump–diffusion process with Gilpin–Ayala drift. The main feature of the Hawkes process is to capture the phenomenon of clustering. The price process is of the mean-reverting type. We prove a sufficient maximum principle for the optimal control of a stochastic system consisting of an SDE driven by the Hawkes process and, by the concavity of the Hamiltonian, we obtain the optimal effort of the fisherman for a risk-averse investor.