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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.

We consider the problem of tracking a Brownian motion by a process of bounded variation, in such a way as to minimize total expected cost of both ‘action' and ‘deviation from a target state 0'. The former is proportional to the amount of control exerted to date, while the latter is being measured by a function which can be viewed, for simplicity, as quadratic. We discuss the discounted, stationary and finite-horizon variants of the problem. The answer to all three questions takes the form of exerting control in a singular manner, in order not to exit from a certain region. Explicit solutions are found for the first and second questions, while the third is reduced to an appropriate optimal stopping problem. This reduction yields properties, as well as global upper and lower bounds, for the associated moving boundary. The pertinent Abelian and ergodic relationships for the corresponding value functions are also derived.

The stochastic control problem of tracking a Brownian motion by a process of bounded variation is reduced to a control problem with reflection at the origin, and the latter is related to a question of optimal stopping of Brownian motion absorbed at the origin. Direct probabilistic arguments can be used to show equivalences between the various problems.