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

TL;DR

This paper characterizes the optimal control in multidimensional singular stochastic control problems as solutions to a Skorokhod reflection problem, providing conditions for their unique identification and applying to linear-quadratic models.

Contribution

It introduces sufficient conditions for the characterization of optimal controls via Skorokhod problems in multidimensional settings, extending previous partial results.

Findings

Optimal control acts only at the boundary of the waiting region.

The direction of control is determined by the derivative of the value function.

The approach applies to both degenerate and non-degenerate models.

Abstract

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 drift of the state equation 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.