  By combining the two-particle-irreducible (2PI) effective action common in
non-equilibrium quantum field theory with the classical Martin-Siggia-Rose
formalism, self-consistent equations of motion for the first and second
cumulants of non-linear classical stochastic processes are constructed. Such
dynamical equations for correlation and response functions are important in
describing non-equilibrium systems, where equilibrium fluctuation-dissipation
relations are unavailable. The method allows to evolve stochastic systems from
arbitrary Gaussian initial conditions. In the non-linear case, it is found that
the resulting integro-differential equations can be solved with considerably
reduced computational effort compared to state-of-the-art stochastic
Runge-Kutta methods. The details of the method are illustrated by several
physical examples.
