  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 It\^o-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.
