A Fermion Levy theorem

The notion of quantum process with continuous trajectories is defined in terms of mutual quadratic variations and it is proved that for classical stochastic processes, this notion of continuity of trajectories coincides with the usual one. Our main result is that any continuous trajectory difference martingale M which is a Grassmann measure with scalar non-atomic brackets is isomorphic to a Fermion white noise (mean zero Fermi-Gaussian family) whose covariance coincides with the brackets of M. This is a fermion version of the Levy representation theorem for classical Brownian motion.