Stationary control of Brownian motion in several dimensions

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 ℝd. 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.