  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.
