High-degree cubature on Wiener space through unshuffle expansions

TL;DR

This paper introduces a new method for constructing high-degree cubature formulas on Wiener space using unshuffle expansions, resulting in more efficient formulas with smaller support.

Contribution

It presents the first explicit degree-7 cubature formula on Wiener space with drift, leveraging Hopf algebra structures for improved efficiency.

Findings

First explicit degree-7 cubature formula on Wiener space with drift

Support of the new formula is significantly smaller than existing methods

Demonstrates effectiveness of unshuffle expansion approach

Abstract

Utilising classical results on the structure of Hopf algebras, we develop a novel approach for the construction of cubature formulae on Wiener space based on unshuffle expansions. We demonstrate the effectiveness of this approach by constructing the first explicit degree-7 cubature formula on $d$-dimensional Wiener space with drift, in the sense of Lyons and Victoir. The support of our degree-7 formula is significantly smaller than that of currently implemented or proposed constructions.