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Stationary control of Brownian motion in several dimensions

Published online by Cambridge University Press:  01 July 2016

R. Mitchell Cox  and
Ioannis Karatzas
R. Mitchell Cox*
Affiliation:
Columbia University
Ioannis Karatzas*
Affiliation:
Columbia University
*
∗Postal address: Department of Mathematical Statistics, Columbia University, New York, NY 10027, USA.
∗Postal address: Department of Mathematical Statistics, Columbia University, New York, NY 10027, USA.
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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 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.

Keywords

STOCHASTIC CONTROLERGODIC PROPERTY OF DIFFUSIONSBESSEL PROCESS WITH DRIFTBELLMAN EQUATIONMAXIMUM PRINCIPLECOMPARISON THEOREMS
Type
Research Article
Information
Advances in Applied Probability , Volume 17 , Issue 3 , September 1985 , pp. 531 - 561
Copyright
Copyright © Applied Probability Trust 1985 

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Footnotes

Research supported in part by the National Science Foundation under grant NSF MCS-81-03435-A01.

Paper presented at the 3rd Bad Honnef Workshop on Stochastic Differential Systems, University of Bonn, FRG, 3–7 June 1985.

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