Optimal ergodic harvesting under ambiguity

TL;DR

This paper addresses an ergodic harvesting problem under model ambiguity by formulating it as a stochastic game, deriving a threshold-based optimal strategy through a free-boundary problem, and analyzing how ambiguity affects the solution.

Contribution

It introduces a novel approach to ergodic harvesting under ambiguity, establishing a threshold policy and resolving a gap in previous HJB analyses.

Findings

Optimal threshold policy derived from free-boundary problem

Optimal payoff depends on ambiguity parameter

As ambiguity vanishes, solution converges to risk-neutral case

Abstract

We consider an ergodic harvesting problem with model ambiguity that arises from biology. To account for the ambiguity, the problem is constructed as a stochastic game with two players: the decision-maker (DM) chooses the `best' harvesting policy and an adverse player chooses the `worst' probability measure. The main result is establishing an optimal strategy (also referred to as a control) of the DM and showing that it is a threshold policy. The optimal threshold and the optimal payoff are obtained by solving a free-boundary problem emerging from the Hamilton--Jacobi--Bellman (HJB) equation. As part of the proof, we fix a gap that appeared in the HJB analysis of [Alvarez and Hening, {\em Stochastic Process. Appl.}, 2019], a paper that analyzed the risk-neutral version of the ergodic harvesting problem. Finally, we study the dependence of the optimal threshold and the optimal payoff on the ambiguity parameter and show that if the ambiguity goes to 0, the problem converges, to the risk-neutral problem.