The Uniqueness of Signature Problem in the Non-Markov Setting

Abstract

The goal of this paper is to simplify and strengthen the Le Jan-Qian approximation scheme of studying the uniqueness of signature problem to the non-Markov setting. We establish a general framework for a class of multidimensional stochastic processes over [0,1] under which with probability one, the signature (the collection of iterated path integrals in the sense of rough paths) is well-defined and determines the sample paths of the process up to reparametrization. In particular, by using the Malliavin calculus we show that our method applies to a class of Gaussian processes including fractional Brownian motion with Hurst parameter H>1/4, the Ornstein-Uhlenbeck process and the Brownian bridge.

31 pages, 1 figure

References in corpus (3)

• Smoothness of the density for solutions to Gaussian rough differential equations
• On an Inversion Theorem for Stratonovich's Signatures of Multidimensional Diffusion Paths
• Simple Piecewise Geodesic Interpolation of Simple and Jordan Curves with Applications

Cited by in corpus (1)

• The Signature of a Rough Path: Uniqueness