We characterize the optimal control for a class of singular stochastic control
problems as the unique solution to a related Skorokhod reflection problem. The considered
optimization problems concern the minimization of a discounted cost functional over an infinite time-horizon through a process of bounded variation affecting an Itô-diffusion. The
setting is multidimensional, the dynamics of the state and the costs are convex, the volatility
matrix can be constant or linear in the state. We prove that the optimal control acts only
when the underlying diffusion attempts to exit the so-called waiting region, and that the
direction of this action is prescribed by the derivative of the value function. Our approach is
based on the study of a suitable monotonicity property of the derivative of the value function
through its interpretation as the value of an optimal stopping game. Such a monotonicity allows to construct nearly optimal policies which reflect the underlying diffusion at the
boundary of approximating waiting regions. The limit of this approximation scheme then
provides the desired characterization. Our result applies to a relevant class of linear-quadratic
models, among others. Furthermore, it allows to construct the optimal control in degenerate
and non degenerate settings considered in the literature, where this important aspect was
only partially addressed.
