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 this paper is to reconstruct a rough path from its signature in an explicit and universal way.