An application of the multiplicative Sewing Lemma to the high order weak approximation of stochastic differential equations

TL;DR

This paper develops a new variant of the multiplicative Sewing Lemma to achieve high-order weak approximations for stochastic differential equations, extending existing cubature methods and providing stability and convergence insights.

Contribution

It introduces a novel variant of the multiplicative Sewing Lemma enabling high-order weak approximations, extending cubature methods on Wiener space.

Findings

Derived stability estimates for the new approximation method.

Established explicit weak convergence rates.

Provided a cubature approximation example for Gaussian martingale-driven SDEs.

Abstract

We introduce a variant of the multiplicative Sewing Lemma in [Gerasimovi\v{c}s, Hocquet, Nilssen; J. Funct. Anal. 281 (2021)] which yields arbitrary high order weak approximations to stochastic differential equations, extending the cubature approximation on Wiener space introduced by Lyons and Victoir. Our analysis allows to derive stability estimates and explicit weak convergence rates. As a particular example, a cubature approximation for stochastic differential equations driven by continuous Gaussian martingales is given.