The functional integral method for the statistical solution of stochastic differential equations is extended to a broad, new class of nonlinear dynamical equations with random coefficients and initial conditions. This work encompasses previous results for classical systems with random forces and initial conditions with arbitrary statistics and provides new results for systems with nonlinear interactions which are nonlocal in time. Closed equations of motion for the correlation and response functions are derived which have applications in the calculation of particle motion in stochastic magnetic fields, in the solution of stochastic wave equations, and in the description of electromagnetic plasma turbulence. As an illustration of the new results for nonlocal interactions, the electromagnetic dispersion tensor is calculated to first order in renormalized theory.