An isomorphism between branched and geometric rough paths

Boedihardjo, H. and Chevyrev, I. (2019) An isomorphism between branched and geometric rough paths. Annales de l'Institut Henri Poincare (B) Probability and Statistics, 55 (2). pp. 1131-1148. ISSN 0246-0203

Abstract/Summary

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	Article
URI	https://reading-clone.eprints-hosting.org/id/eprint/76834
Refereed	Yes
Divisions	Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics
Publisher	Institute Henri Poincaré
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