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The traditional difficulty about stochastic singular control is tocharacterize the regularities of the value function and the optimal controlpolicy. In this paper, a multi-dimensional singular control problem isconsidered. We found the optimal value function and the optimal control policyof this problem via Dynkin game, whose solution is given by the saddle point ofthe cost function. The existence and uniqueness of the solution to this Dynkingame are proved through an associated variational inequality problem involvingDirichlet form. As a consequence, the properties of the value function of thisDynkin game implies the smoothness of the value function of the stochasticsingular control problem. In this way, we are able to show the existence of aclassical solution to this multi-dimensional singular control problem, whichwas traditionally solved in the sense of viscosity solutions, and this enablesthe application of the verification theorem to prove optimality.

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