We establish a general framework for a class of multidimensional stochastic processes over 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 , the Ornstein–Uhlenbeck process and the Brownian bridge.