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We characterize the optimal control for a class of singular stochasticcontrol problems as the unique solution to a related Skorokhod reflectionproblem. The considered optimization problems concern the minimization of adiscounted cost functional over an infinite time-horizon through a process ofbounded variation affecting an It\^o-diffusion. The setting ismultidimensional, the drift of the state equation and the costs are convex, thevolatility matrix can be constant or linear in the state. We prove that theoptimal control acts only when the underlying diffusion attempts to exit theso-called waiting region, and that the direction of this action is prescribedby the derivative of the value function. Our approach is based on the study ofa suitable monotonicity property of the derivative of the value functionthrough its interpretation as the value of an optimal stopping game. Such amonotonicity allows to construct nearly optimal policies which reflect theunderlying diffusion at the boundary of approximating waiting regions. Thelimit of this approximation scheme then provides the desired characterization.Our result applies to a relevant class of linear-quadratic models, amongothers. Furthermore, it allows to construct the optimal control in degenerateand non degenerate settings considered in the literature, where this importantaspect was only partially addressed.

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