High-degree cubature on Wiener space through unshuffle expansions
Emilio Ferrucci, Timothy Herschell, Christian Litterer, Terry Lyons
TL;DR
High-degree cubature on Wiener space with drift is challenging in high dimensions due to large moment systems. The authors introduce a Hopf-algebraic unshuffle expansion that uses a redundant spanning set given by the Eulerian idempotent to express the expected signature, turning the moment problem into a sparser set of constraints. They construct an explicit degree-$7$ cubature formula in arbitrary dimension by modeling the Lie-polynomial coefficients with products of Gaussian variables and realizing them via Gaussian cubatures for the auxiliary variables. This yields cubature measures with substantially smaller support than existing constructions, improving efficiency for the Lyons–Victoir/KLV framework and enabling scalable high-dimensional weak approximations.
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.