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

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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
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of 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:	Date Event 14 May 2019 Published 24 April 2018 Accepted
Date of first compliant deposit:	5 July 2020
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
Related URLs:	Publisher
Open Access Version:	ArXiv

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