  We discuss general multi-dimensional stochastic processes driven by a system
of Langevin equations with multiplicative white noise. In particular, we
address the problem of how time reversal diffusion processes are affected by
the variety of conventions available to deal with stochastic integrals. We
present a functional formalism to built up the generating functional of
correlation functions without any type of discretization of the Langevin
equations at any intermediate step. The generating functional is characterized
by a functional integration over two sets of commuting variables as well as
Grassmann variables. In this representation, time reversal transformation
became a linear transformation in the extended variables, simplifying in this
way the complexity introduced by the mixture of prescriptions and the
associated calculus rules. The stochastic calculus is codified in our formalism
in the structure of the Grassmann algebra. We study some examples such as
higher order derivatives Langevin equations and the functional representation
of the micromagnetic stochastic Landau-Lifshitz-Gilbert equation.
