Consider the stochastic differential equation on $\mathbb R$
$$ dx_t = f(x_t) dt + g(\omega t)\, dW_t $$
with $W_t$ a standard Brownian motion, $f:\mathbb R \to \mathbb R$ a smooth function, and $g:\mathbb R\to \mathbb R$ a 1-periodic function. Let $c^2 := \int_0^1 g^2(s)ds$ be the square of the average of $g$. I would like to show that the solutions to the SDE above converge in a probabilistic sense to the solution of $$ dy_t = f(y_t) dt + c\, dW_t $$ over each finite interval of time $[0,T]$ in the limit $\omega \to \infty$. In other words, I would like to show that if the diffusion coefficient oscillates very rapidly, then it can essentially be replaced with its root mean square.
This must somehow follow from Levi's theorem, but it is not clear to me how. I guess this is a classical exercice, but I fail to find it anywhere... Any proof or reference would be much appreciated.