# Wong-Zakai smooth approximation in probabilty for stochastic differential equations

I'm looking for a result of the form: Let $B_\epsilon$ denote a "natural" smooth $\epsilon$-approximation to an $n$-dimensional Brownian motion $B$ (e.g. by mollification or simply piecewise linear) with $\sigma$ and $\mu$ smooth coefficients, and denote by $X_\epsilon$ the solution to the random ODE $$dX_\epsilon = \mu(X_\epsilon)dt + \sigma(X_\epsilon)dB_\epsilon.$$

Then $X_\epsilon$ converges in probability, as $\epsilon \to 0$, so the solution of the SDE $$dX=\mu(X)dt+\sigma(X)\circ dB.$$

In particular I'm seeking results with no ellipticity requirement on $\sigma$. Many thanks.

• The original Wong-Zakai paper actually doesn't treat the $n$-dimensional case. In the one-dimensional case, any (continuous) approximation converges, which is not the case in higher dimensions. To the best of my knowledge the first articles treating the $n$-dimensional case are Strook & Varadhan's support theorem paper. (Coincidentally, there's also a paper by McShane with a similar result that appeared in the same proceedings volume.) – Martin Hairer Feb 7 '16 at 14:07