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Optimally Coupling the Kolmogorov Diffusion, and Related Optimal Control Problems

Published online by Cambridge University Press:  01 February 2010

Kalvis M. Jansons  and
Paul D. Metcalfe
Kalvis M. Jansons
Affiliation:
Department of Mathematics, University College London, Gower Street, London WC1E 6BT, United Kingdom, [email protected], http://www.kalvis.com
Paul D. Metcalfe
Affiliation:
Cyprotex Discovery Ltd., 15 Beech Lane, Macclesfield SK10 2DR, United Kingdom

Abstract

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We discuss the optimal Markovian coupling before an exponential time of the Kolmogorov diffusion, and a class of related stochastic control problems in which the aim is to hit the origin before an exponential time. We provide a scaling argument for the optimal control in the near field and use rational WKB approximation to obtain the optimal control in the far field, and compare these analytical results with numerical experiments. In some of these optimal control problems, in which the advection velocity field is bounded, we show that the probability of success agrees exactly with its leading-order asymptotic approximation in some areas of the plane, up to an undetermined multiplicative constant. We conjecture a necessary and sufficient condition for this behaviour, which is strongly supported by numerical experiments.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 10 , 2007 , pp. 1 - 20
Copyright
Copyright © London Mathematical Society 2007

References

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