Reconstruction for the Signature of a Rough Path

Abstract

Recently it was proved that the group of rough paths modulo tree-like equivalence is isomorphic to the corresponding signature group through the signature map S (a generalized notion of taking iterated path integrals). However, the proof of this uniqueness result does not contain any information on how to "see" the trajectory of a (tree-reduced) rough path from its signature, and a constructive understanding on the uniqueness result (in particular on the inverse of S) has become an interesting and important question. The aim of the present paper is to reconstruct a rough path from its signature in an explicit and universal way.

39 pages, 6 figures

References in corpus (6)

• A theory of regularity structures
• Uniqueness for the signature of a path of bounded variation and the reduced path group
• Learning from the past, predicting the statistics for the future, learning an evolving system
• Vanishing of one dimensional L^2-cohomologies of loop groups
• Inverting the signature of a path
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• SL_2(R)-developments and Signature Asymptotics for Planar Paths with Bounded Variation