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

*Hidekazu Yoshioka* and
*Yuta Yaegashi*

*Mathematics and Computers in Simulation (MATCOM)*, 2019, vol. 156, issue C, 40-66

**Abstract:**
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.

**Keywords:** Finite difference scheme; Singular stochastic control; Variational inequality; Free boundary; Stochastic differential game (search for similar items in EconPapers)

**Date:** 2019

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**Persistent link:** https://EconPapers.repec.org/RePEc:eee:matcom:v:156:y:2019:i:c:p:40-66

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