  Multiplicative white-noise stochastic processes continuously attract the
attention of a wide area of scientific research. The variety of prescriptions
available to define it difficults the development of general tools for its
characterization. In this work, we study equilibrium properties of Markovian
multiplicative white-noise processes. For this, we define the time reversal
transformation for this kind of processes, taking into account that the
asymptotic stationary probability distribution depends on the prescription.
Representing the stochastic process in a functional Grassman formalism, we
avoid the necessity of fixing a particular prescription. In this framework, we
analyze equilibrium properties and study hidden symmetries of the process. We
show that, using a careful definition of equilibrium distribution and taken
into account the appropriate time reversal transformation, usual equilibrium
properties are satisfied for any prescription. Finally, we present a detailed
deduction of a covariant supersymmetric formulation of a multiplicative
Markovian white-noise process and study some of the constraints it imposes on
correlation functions using Ward-Takahashi identities.
