Note: This is a strengthening of the following result, motivated by the need for strong convergence in applications.

Let $W$ be a one dimensional standard Brownian motion, and let $X$ be the solution to the SDE

$$dX_t = \sigma(X_t) \, dW_t \, , \, X_0 = 0$$

with $\sigma: \mathbb R \to \mathbb R$ Lipschitz continuous.

For each $c > 0$, define the process $Y^c$ on $[0, 1]$ by

$$X^c_t := c^{-1/2} X_{ct}.$$

Similarly define

$$W^c_t := c^{-1/2} \, W_{ct}.$$

Question: Is it true that as $c \to 0^+$, we have

$$\mathbb E[\sup_{0 \leq t \leq 1} |X^c_t - \sigma(0) W^c_t|] \to 0?$$