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Coupling the Kolmogorov diffusion: maximality and efficiency considerations

Part of: Stochastic processes Markov processes

Published online by Cambridge University Press:  25 July 2016

Sayan Banerjee  and
Wilfrid S. Kendall
Sayan Banerjee*
Affiliation:
University of Warwick
Wilfrid S. Kendall*
Affiliation:
University of Warwick
*
Department of Statistics, University of Warwick, Coventry CV4 7AL, UK. Email address: sayan.banerjee@warwick.ac.uk
Department of Statistics, University of Warwick, Coventry CV4 7AL, UK. Email address: w.s.kendall@warwick.ac.uk
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Abstract

This is a case study concerning the rate at which probabilistic coupling occurs for nilpotent diffusions. We focus on the simplest case of Kolmogorov diffusion (Brownian motion together with its time integral or, more generally, together with a finite number of iterated time integrals). We show that in this case there can be no Markovian maximal coupling. Indeed, there can be no efficient Markovian coupling strategy (efficient for all pairs of distinct starting values), where the notion of efficiency extends the terminology of Burdzy and Kendall (2000). Finally, at least in the classical case of a single time integral, it is not possible to choose a Markovian coupling that is optimal in the sense of simultaneously minimizing the probability of failing to couple by time t for all positive t. In recompense for all these negative results, we exhibit a simple efficient non-Markovian coupling strategy.

Keywords

Brownian motionBrownian time integralco-adapted couplingcouplingefficient couplingfiltrationfinite-look-ahead couplinghypoelliptic diffusionimmersed coupling; Karhunen‒Loève expansionKolmogorov diffusionMarkovian couplingmaximal couplingnilpotent diffusionoptimal Markovian couplingreflection couplingsynchronous coupling

MSC classification

Primary: 60G05: Foundations of stochastic processes
Secondary: 60J60: Diffusion processes
Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue A: Probability, Analysis and Number Theory , July 2016 , pp. 15 - 35
Copyright
Copyright © Applied Probability Trust 2016 

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