Optimally coupling the Kolmogorov diffusion, and related optimal control problems
Kalvis M. Jansons and Paul D. Metcalfe
Abstract: The authors 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. They 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, they 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. They conjecture a
necessary and sufficient condition for this behaviour, which is
strongly supported by numerical experiments.
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