Optimal multi-dimensional stochastic harvesting with density-dependent prices

TL;DR

This paper develops a verification theorem for complex stochastic control problems involving density-dependent prices, providing explicit solutions and revealing the existence of chattering policies when prices decrease with population density.

Contribution

It introduces a verification theorem for singular control problems with density-dependent prices and explicitly characterizes optimal policies, including the novel concept of chattering policies.

Findings

Optimal harvesting policies may not exist in the traditional sense when prices decrease with density.

Chattering policies can be used as limits of small, rapid harvesting actions.

Explicit solutions are provided for specific examples of such control problems.

Abstract

We prove a verification theorem for a class of singular control problems which model optimal harvesting with density-dependent prices or optimal dividend policy with capital-dependent utilities. The result is applied to solve explicitly some examples of such optimal harvesting/optimal dividend problems. In particular, we show that if the unit price decreases with population density, then the optimal harvesting policy may not exist in the ordinary sense, but can be expressed as a "chattering policy", i.e. the limit as $\Delta x$ and $\Delta t$ go to $0$ of taking out a sequence of small quantities of size $\Delta x$ within small time periods of size $\Delta t$.