We study the role of multiplicative stochastic processes in the description of the dynamics of an order parameter near a critical point. We study equilibrium, as well as, out-of-equilibrium properties. By means of a functional formalism, we built the Dynamical Renormalization Group equations for a real scalar order parameter with $Z_2$ symmetry, driven by a class o multiplicative stochastic processes with the same symmetry. We have computed the flux diagram, using a controlled $ε$-expansion, up to order $ε^2$. We have found that, for dimensions $d=4-ε$, the additive dynamic fixed point is unstable. The flux runs to a {\em multiplicative fixed point} driven by a diffusion function $G(ϕ)=1+g^*ϕ^2({\bf x})/2$, where $ϕ$ is the order parameter and $g^*=ε^2/18$ is the fixed point value of the multiplicative noise coupling constant. We show that, even though the position of the fixed point depends on the stochastic prescription, the critical exponents do not. Therefore, different dynamics driven by different stochastic prescriptions (such as Itô, Stratonovich, anti-Itô and so on) are in the same universality class.

34 pages, 14 figures. Extended version. Final version accepted for publication in Physica A: Statistical Mechanics and its Applications