Writing about web page https://arxiv.org/abs/1705.01600
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 ∫Wi1Wj2∘dW2, where j=1,…,n and i=0,…,n−j 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."