# Coupling Polynomial Stratonovich Integrals: the two-dimensional Brownian case

## TL;DR

This paper develops a method to construct an immersion coupling of a two-dimensional Brownian motion with multiple polynomial Stratonovich integrals, advancing the understanding of coupling complex stochastic functionals.

## Contribution

It introduces a novel coupling construction for Brownian motion and its polynomial integrals, extending previous work on stochastic area and time integral couplings.

## Key findings

- Constructed explicit couplings for polynomial integrals of Brownian motion.
- Applied the coupling method to hypoelliptic diffusions driven by polynomial vector fields.
- Enhanced understanding of when multiple integral functionals can be coupled.

## Abstract

We show how to build an immersion coupling of a two-dimensional Brownian motion $(W_1, W_2)$ along with $\binom{n}{2} + n= \tfrac12n(n+1)$ integrals of the form $\int W_1^iW_2^j \circ dW_2$, where $j=1,\ldots,n$ and $i=0, \ldots, n-j$ 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 multiple selected integral functionals of the diffusions.