High-degree cubature on Wiener space through unshuffle expansions

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Abstract
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.
Funding Information
  • Alan Turing Institute: the Defence and Security Programme
  • Crankstart Internship Programme
  • Hong Kong Innovation and Technology Commissio (IONNOHK CIMDA (Oxford))
  • Engineering and Physical Sciences Research Council (EP/S026347/1)
  • Engineering and Physical Sciences Research Council (EP/V005413/1)
  • Engineering and Physical Sciences Research Council (UKRI1010)
  • Engineering and Physical Sciences Research Council (UKRI2385)

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