Actually, I think the random function CAN be defined numerically in advance. Since
is statistically the same as
I can write my set of ODE's as:
where Q[t]=Random[NormalDistribution[0,1]] at each instant in time.
I generated Q[t] in advance by creating a table of such random numbers at many time values, and then invoking Interpolation[...]. The solutions generated look just like those in the attachement of the above reply, but NDSolve only needs to be called once. See this attached notebook.
Attachment: StochasticODE.nb, URL: ,