April 19, 2016

Coupling the Kolmogorov Diffusion: maximality and efficiency considerations

Writing about web page https://arxiv.org/abs/1506.04804

Banerjee, S., & Kendall, W. S. (2016). Coupling the Kolmogorov Diffusion: maximality and efficiency considerations. Advances in Applied Probability, 48A, to appear. See https://arxiv.org/abs/1506.04804 p, li { white-space: pre-wra

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, slightly more generally, together with a finite number of iterated time integrals). In this case there can be no Markovian maximal coupling. Indeed, Markovian couplings cannot even be efficient (extending the terminology of Burdzy & Kendall, Efficient Markovian couplings: examples and counterexamples; Annals of Applied Probability, 2000, 10.2, 362–409). 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.


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