  We introduce a new methodology based on the multirevolution idea for
constructing integrators for stochastic differential equations in the situation
where the fast oscillations themselves are driven by a Stratonovich noise.
Applications include in particular highly-oscillatory Kubo oscillators and
spatial discretizations of the nonlinear Schr\"odinger equation with fast white
noise dispersion. We construct a method of weak order two with computational
cost and accuracy both independent of the stiffness of the oscillations. A
geometric modification that conserves exactly quadratic invariants is also
presented.
