Wasserstein approximations of the L\'evy area random walk via polynomial perturbations of Gaussian distributions

TL;DR

This paper develops a new coupling method to approximate the Le9vy area increments of Brownian motion using polynomial perturbations of Gaussian distributions, improving pathwise approximation schemes for stochastic differential equations.

Contribution

It introduces a novel coupling construction based on Wasserstein estimates and polynomial perturbations, advancing the approximation of Le9vy area in stochastic analysis.

Findings

Constructs a coupling between Le9vy area increments and polynomial Gaussian perturbations.

Provides Wasserstein estimates for the approximation error.

Enhances pathwise approximation schemes for SDEs.

Abstract

We construct a coupling between the random walk composed of L\'evy area increments from a $d$-dimensional Brownian motion and a random walk composed of quadratic polynomials of Gaussian random variables. This coupling construction is used to produce a new pathwise approximation scheme for stochastic differential equations in the preprint [Flint-Lyons-2015]. The coupling arguments of the present paper are based extensively on the recent coupling results of Davie concerning a multidimensional variant of the Koml\'os-Major-Tusn\'ady theorem and Wasserstein estimates for polynomial perturbations of Gaussian measures.