A finite difference scheme for variational inequalities arising in stochastic control problems with several singular control variables

摘要

A finite difference scheme is developed for solving 1-D variational inequalities arising in stochastic control problems with several singular control variables. The scheme guarantees the uniqueness of numerical solutions. A policy iteration algorithm is then proposed to solve the discretized problem. The present approach is applied to solving variational inequalities associated to cost-effective management problems of benthic algae on the riverbed downstream of a dam: an urgent environmental problem. Accuracy of the scheme is verified to be first-order for both the solution and its free boundaries. An advanced problem that involves a max-min differential game structure is also examined. The scheme then computes reasonably accurate numerical solutions which are consistent with the theoretical asymptotic estimates. (C) 2018 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.