Ito diffusion with highly oscillatory diffusion coefficient

Question

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.

Answer 1

This is true, but the limiting process is driven by a Brownian motion $B$ which is different from $W$. To prove this, use first the Dambis-Dubins-Schwarz representation of a continuous martingale (see the book by Revuz & Yor for example) to see that $A_\omega(\cdot) = \int_0^\cdot g(\omega t)\,dW$ is a time-change of a Brownian motion $B$. The time change in question is given by the integral of $g^2(\omega t)$, which converges to $c^2t$ so that, by Hölder continuity of $B$, $A_\omega$ converges weakly in the space of continuous functions as $\omega \to \infty$ to some Wiener process $B$. The proof is concluded by observing that the map that maps $A$ to the solution to the integral equation $$ y_t = y_0 + \int_0^t f(y_s)\,ds + A(t) $$ is continuous in the topology of continuous functions.

The usual caveats apply, i.e. $f$ needs to be either globally Lipschitz or at least have enough dissipativity, otherwise convergence only holds up to a possible blow-up time. (And the proof gets slightly more technical.) As a bonus exercise, show that the pair $(W,A)$ converges jointly to two Brownian motions $(W,B)$ with a non-trivial covariance matrix and compute that covariance...

Answer 2

Another approach is to try to use Theorem 4.1 from the section on diffusion approximation of the book of Ethier and Kurtz on Markov processes (and their convergence). Of course, this will give weak convergence of distributions whereas Martin's approach is more pathwise.