Multidimensional singular control and related Skorokhod problem: Sufficient conditions for the characterization of optimal controls

We characterize the optimal control for a class of singular stochastic control problems as the unique solution to a related Skorokhod reflection problem. The optimization problems concern the minimization of a discounted cost over an infinite time-horizon through a process of bounded variation affecting an Itô-diffusion. The setting is multidimensional, the drift of the state equation and the costs are convex, the volatility matrix can be constant or linear in the state. Our result applies to a relevant class of linear-quadratic models and it allows to construct the optimal control in degenerate and non degenerate settings considered in the literature.