It has been a while since I studied along these lines (stochastic chemical reaction simulations in my case) but I think the answer is often yes, but not always (I don't think). A random walk for example will be a normal distribution (and you know the mean, and you know the variance is going to infinity), so I do think in that case you end up with an elegant analytical solution if I'm understanding correctly as the inputs can determine the function the variance follows through time.

But often no, you need to run a stochastic algorithm (e.g. Gillespie's algorithm in the case of simple stochastic chemical kinetics) as there will be no analytical solution.

Again it has been a while though.