Stratonovich's Signatures of Brownian Motion Determine Brownian Sample Paths

TL;DR

This paper proves that in dimensions two and higher, the entire sample path of a Brownian motion can be uniquely reconstructed from its Stratonovich signature, highlighting the signature's power as a complete path invariant.

Contribution

It establishes that Brownian motion paths in dimensions ≥2 are uniquely determined by their signatures, extending the understanding of signatures as complete path descriptors.

Findings

Brownian paths are almost surely determined by their signatures in dimensions ≥2

Signature captures all information of the Brownian sample paths

Advances the theory of path signatures in stochastic analysis

Abstract

The signature of Brownian motion in $\mathbb{R}^{d}$ over a running time interval $[0,T]$ is the collection of all iterated Stratonovich path integrals along the Brownian motion. We show that, in dimension $d\geq 2$, almost all Brownian motion sample paths (running up to time $T$) are determined by its signature over $[0,T]$