Signature inversion for monotone paths

TL;DR

This paper introduces a sampling method to reconstruct monotone paths from their signatures, demonstrating that the probability of incorrect reconstruction diminishes exponentially with the number of steps, based on large deviations principles.

Contribution

It provides a novel sampling procedure for reconstructing monotone paths from signatures and establishes large deviations results for the sampling weights.

Findings

Sampling weights satisfy large deviations principle

Probability of wrong path selection is exponentially small

Probabilistic interpretation of signature for monotone paths

Abstract

The aim of this article is to provide a simple sampling procedure to reconstruct any monotone path from its signature. For every N, we sample a lattice path of N steps with weights given by the coefficient of the corresponding word in the signature. We show that these weights on lattice paths satisfy the large deviations principle. In particular, this implies that the probability of picking up a "wrong" path is exponentially small in N. The argument relies on a probabilistic interpretation of the signature for monotone paths.