Stationary control of Brownian motion in several dimensions

Abstract

We address the question of controlling the Brownian path in several dimensions (d≧2) by continually choosing its drift from among vectors of the unit ball in ℝ^d. The past and present of the path are supposed to be completely observable, while no anticipation of the future is allowed. Imposing a suitable cost on distance from the origin, as well as a cost of effort proportional to the length of the drift vector, ‘reasonable’ procedures turn out to be of the following type: to apply drift of maximal length along the ray towards the origin if the current position is outside a sphere centred at the origin, and to choose zero drift otherwise. It is shown just how to compute the radius of such a sphere in terms of the data of the problem, so that the resulting procedure is optimal.