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An isomorphism between branched and geometric rough paths
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Boedihardjo, Horatio and Chevyrev, Ilya (2019) An isomorphism between branched and geometric rough paths. Annales de l'Institut Henri Poincare (B) Probability and Statistics, 55 (2). pp. 1131-1148. doi:10.1214/18-AIHP912 ISSN 0246-0203 .
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Official URL: https://doi.org/10.1214/18-AIHP912
Abstract
We exhibit an explicit natural isomorphism between spaces of branched and geometric rough paths. This provides a multi-level generalisation of the isomorphism of Lejay-Victoir as well as a canonical version of the Itô-Stratonovich correction formula of Hairer-Kelly. Our construction is elementary and uses the property that the Grossman-Larson algebra is isomorphic to a tensor algebra. We apply this isomorphism to study signatures of branched rough paths.Namely, we show that the signature of a branched rough path is trivial if and only if the path is tree-like, and construct a non-commutative Fourier transform for probability measures on signatures of branched rough paths. We use the latter to provide sufficient conditions for a random signature to be determined by its expected value, thus giving an answer to the uniqueness moment problem for branched rough paths.
Item Type: | Journal Article | ||||||
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Subjects: | Q Science > QA Mathematics | ||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Statistics | ||||||
Journal or Publication Title: | Annales de l'Institut Henri Poincare (B) Probability and Statistics | ||||||
Publisher: | Institute Henri Poincaré | ||||||
ISSN: | 0246-0203 | ||||||
Official Date: | 14 May 2019 | ||||||
Dates: |
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Volume: | 55 | ||||||
Number: | 2 | ||||||
Page Range: | pp. 1131-1148 | ||||||
DOI: | 10.1214/18-AIHP912 | ||||||
Status: | Peer Reviewed | ||||||
Publication Status: | Published | ||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||
Date of first compliant deposit: | 5 July 2020 | ||||||
Date of first compliant Open Access: | 6 July 2020 | ||||||
Related URLs: | |||||||
Open Access Version: |
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