I know that in order to solve a differential equation with white noise a stochastic numerical integration technique must be used. As I understand it, this is because the white noise has zero correlation time and therefore has unbounded variation. My question is supposing we have noise with a non-zero correlation time, under what conditions is it a good approximation to use standard (non stochastic) numerical integration techniques? To be more concrete, suppose I have an equation of the form $$dy = f(y) dt + g(y)z dt $$ $$dz = -a z dt + b dW $$ where $z$ is colored noise (the Ornstein-Uhlenbeck process) with the properties $$\left< z(t) \right> = 0 ~~~~~~~\text{and}~~~~~~ \left< z(t) z(s) \right> = \frac{b^2}{2a}\exp(-a|t-s|)$$ where $\tau = a^{-1}$ is the correlation time. I know that the equation $dz = ...$ needs to be integrated using a stochastic integration method. The question is whether $dy = ...$ can be integrated using a deterministic method once $z_k, z_{k+1}, ...$ is known.
My guess would be that if I integrate using say a standard Runge-Kutta scheme with a time step $dt << \tau$ (say $dt < 5\tau$) then the results would be more or less the same as those obtained using stochastic numerical techniques. Is this correct? Is there some mathematical theorem which proves this?