Finite Difference Scheme for Stochastic Differential Games with Several Singular Control Variables and Its Environmental Application

Abstract

Stochastic differential games have recently been key mathematical tools for resolution of environmental and ecological optimization problems. A finite difference scheme is proposed for solving a variational inequality arising in a stochastic differential game having several singular control variables: optimization of algae population management under model ambiguity. The present scheme employs fitted-exponential and upwind discretization methods to generate stable numerical solutions. Accuracy of the scheme is verified against an exact solution to a simplified problem and an asymptotic solution to a more complicated problem. The scheme is finally applied to numerical computation of the optimal algae population management policy against a range of an incurred cost. The computational results suggest that qualitatively different optimal policies are obtained depending on the magnitude of the incurred cost.