Multi-dimensional Stochastic Singular Control Via Dynkin Game and Dirichlet Form

TL;DR

This paper addresses multi-dimensional stochastic singular control by characterizing the value function and optimal policy through Dynkin games and Dirichlet forms, establishing classical solutions and optimality verification.

Contribution

It introduces a novel approach using Dynkin games and Dirichlet forms to obtain classical solutions for multi-dimensional singular control problems.

Findings

Existence and uniqueness of the Dynkin game solution proved.

Smoothness of the value function established.

Classical solutions enable verification theorem application.

Abstract

The traditional difficulty about stochastic singular control is to characterize the regularities of the value function and the optimal control policy. In this paper, a multi-dimensional singular control problem is considered. We found the optimal value function and the optimal control policy of this problem via Dynkin game, whose solution is given by the saddle point of the cost function. The existence and uniqueness of the solution to this Dynkin game are proved through an associated variational inequality problem involving Dirichlet form. As a consequence, the properties of the value function of this Dynkin game implies the smoothness of the value function of the stochastic singular control problem. In this way, we are able to show the existence of a classical solution to this multi-dimensional singular control problem, which was traditionally solved in the sense of viscosity solutions, and this enables the application of the verification theorem to prove optimality.