Singular Control of Multidimensional Brownian Motion †

Abstract

A multidimensional Brownian motion with an additive input (control) is considered. There is no limit of the rate on input. The cost of control is proportional to the weighted sum of the total amounts of positive and negative input components. There is also a holding cost associated with the position of the process in the space. The problem is to minimize the total discounted holding cost and the cost of control. It is proved that the Hamilton-Jacobi-Bellman equation has a solution. The solution is the optimal cost of the problem. An existence of the optimal policy is demonstrated.