May 29, 2017

Coupling Polynomial Stratonovich Integrals: the two–dimensional Brownian case

Sayan Banerjee and I have recently posted an article on arXiv entitled "Coupling Polynomial Stratonovich Integrals: the two-dimensional Brownian case", which describes rather general couplings for polynomial stochastic integrals driven by two-dimensional Brownian motions. From the abstract:
"We show how to build an immersion coupling of a two-dimensional Brownian motion (W1,W2) along with (n2)+n=12n(n+1) integrals of the form Wi1Wj2dW2, where j=1,,n and i=0,,nj for some fixed n. The resulting construction is applied to the study of couplings of certain hypoelliptic diffusions (driven by two-dimensional Brownian motion using polynomial vector fields). This work follows up previous studies concerning coupling of Brownian stochastic areas and time integrals (Ben Arous, Cranston and Kendall (1995), Kendall and Price (2004), Kendall (2007), Kendall (2009), Kendall (2013), Banerjee and Kendall (2015), Banerjee, Gordina and Mariano (2016)), and is part of an ongoing research programme aimed at gaining a better understanding of when it is possible to couple not only diffusions but also selected integral functionals of the diffusions."

April 19, 2016

Rigidity for Markovian Maximal Couplings of Elliptic Diffusions

Banerjee, S., & Kendall, W. S. (2016). Rigidity for Markovian Maximal Couplings of Elliptic Diffusions. Probab. Theory Related Fields, 42(to appear), 58pp. http://doi.org/10.1007/s00440-016-0706-4

Maximal couplings are (probabilistic) couplings of Markov processes such that the tail probabilities of the coupling time attain the total variation lower bound (Aldous bound) uniformly for all time. Markovian (or immersion) couplings are couplings defined by strategies where neither process is allowed to look into the future of the other before making the next transition. Markovian couplings are typically easier to construct and analyze than general couplings, and play an important role in many branches of probability and analysis. Hsu and Sturm (2013) proved that the reflection-coupling of Brownian motion is the unique Markovian maximal coupling (MMC) of Brownian motions starting from two different points. Later, Kuwada (2009) proved that the existence of a MMC for Brownian motions on a Riemannian manifold enforces existence of a reflection structure on the manifold. In this work, we investigate suitably regular elliptic diffusions on manifolds, and show how consideration of the diffusion geometry (including dimension of the isometry group and flows of isometries) is fundamental in classification of the space and the generator of the diffusion for which an MMC exists, especially when the MMC also holds under local perturbations of the starting points for the coupled diffusions. We also describe such diffusions in terms of Killing vectorfields (generators of isometry groups) and dilation vectorfields (generators of scaling symmetry groups).

Coupling the Kolmogorov Diffusion: maximality and efficiency considerations

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.

July 26, 2013

Talk by WSK at London Probability Seminar (Imperial College), 28 October 2013

This talk will be in the afternoon. In the morning Hermann Thorisson will be giving 2 lectures in the morning to introduce probabilistic coupling, as well as giving a seminar in the afternoon.

My own talk title and abstract:

Title:
Probabilistic coupling and Nilpotent Diffusions
Abstract:
Modern probability theory makes considerable use of the technique of probabilistic coupling. The idea is, to analyse a given random process by constructing two inter-dependent copies of it, defined on the same probability space, and related in such a way as to facilitate analysis. Applications include: establishing monotonicity in non-obvious situations, developing quantitative approximations to distributions of random variables, constructing gradient estimates, and even producing exact simulation algorithms for Markov chains.
However the thematic question, which has driven much of the theory of probabilistic coupling, concerns whether or not one can construct the two coupled random processes so that they almost surely meet ("couple") at some future random time, and if so then whether one can construct a maximal coupling, for which the random time is smallest possible? The question is sharpened if we require the coupling to be co-adapted (also: immersed, or Markovian); this is an additional requirement that the coupling respect the underlying causal structure of the random processes, and can be viewed as implying that the coupling is easily constructable in some general sense.
There is a considerable body of theory describing how to build successful co-adapted couplings for elliptic diffusions, all building on the basic reflection coupling for simple random walks or Brownian motion (very simply, the random jumps of the coupled process are arranged so far as possible to be the opposites of the random jumps of the original process). It is conjectured that successful co-adapted couplings can be built for all hypoelliptic diffusions (diffusions in d dimensions with fewer than d "directions of randomness").
In this talk I will survey the general theory of coupling, describe the known results for co-adapted couplings of hypoelliptic diffusions (in fact, Brownian motions on nilpotent Lie groups), and briefly discuss a related and very simple example which has applications to the theory of filtrations.

May 01, 2013

EPSRC Project Proposal

Here is a PDF of the successful proposal for this EPSRC-funded project. It is placed here for the benefit of anyone interested in applying for the Statistics PDRA post now being advertised.

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