Inversion of signature for paths of bounded variation

TL;DR

This paper introduces two methods for reconstructing paths of bounded variation from their signatures, one explicit for axis paths and another approximation scheme for general paths, advancing the understanding of path signature inversion.

Contribution

It presents novel explicit and approximation-based methods for inverting path signatures, extending the applicability to general bounded variation paths.

Findings

Explicit formula for axis paths from signatures

Approximation scheme for general paths

Convergence of lattice path approximations

Abstract

We develop two methods to reconstruct a path of bounded variation from its signature. The first method gives a simple and explicit expression of any axis path in terms of its signature, but it does not apply directlty to more general ones. The second method, based on an approximation scheme, recovers any tree-reduced path from its signature as the limit of a uniformly convergent sequence of lattice paths.