High-degree cubature on Wiener space through unshuffle expansions

Journal article

Utilizing 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 degree-7 cubature formula on d-dimensional Wiener space with drift in the sense of Lyons and Victoir (Lyons & Victoir 2004 Stoch. Anal. Appl. Math. Finance460, 169–198) which is explicit as a function of an underlying Gaussian cubature. The support of our degree-7 formula is significantly smaller than that of currently implemented or proposed constructions.

Authors: Ferrucci, E; Herschell, T; Litterer, C; Lyons, TJ

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: The Royal Society
• Journal: Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
• Volume: 482
• Issue: 2330
• Pages: 20250051
• Article number: 20250051
• Publication date: 2026-01-21
• Acceptance date: 2025-11-17
• DOI: 10.1098/rspa.2025.0051
• EISSN: 1471-2946
• ISSN: 1364-5021
• Language: English
• Keywords: cubature on Wiener space; Lie idempotents; weak approximation to SDEs; numerical methods for SDEs; Hopf algebras