On the optimally controlled stochastic shallow lake

TL;DR

This paper advances the understanding of the stochastic control problem for shallow lakes by generalizing the value function characterization, establishing approximate optimality, and developing a numerical scheme to analyze properties of optimal control.

Contribution

It generalizes the viscosity solution characterization for broader recycling rates, proves approximate optimality with bounded controls, and introduces a stable numerical method for analysis.

Findings

Derived tail asymptotics for the invariant distribution

Extended results beyond small noise limit

Provided a convergent numerical scheme for the value function

Abstract

We consider the stochastic control problem of the shallow lake and continue the work of G. T. Kossioris, Loulakis, and Souganidis (2019) in three directions. First, we generalise the characterisation of the value function as the viscosity solution of a well-posed problem to include more general recycling rates. Then, we prove approximate optimality under bounded controls and we establish quantitative estimates. Finally, we implement a convergent and stable numerical scheme for the computation of the value function to investigate properties of the optimally controlled stochastic shallow lake. This approach permits to derive tail asymptotics for the invariant distribution and to extend results of Grass, Kiseleva, and Wagener (2015) beyond the small noise limit.