Stratonovich's signatures of Brownian motion determine Brownian sample paths

Abstract

The signature of Brownian motion in ℝd 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≥ 2, almost all Brownian motion sample paths (running up to time T) are determined by their signature over [0,T]. © 2012 Springer-Verlag Berlin Heidelberg.