Let $W$ be a standard one dimensional Brownian motion, and consider the SDE
$$dX_t = \sigma(X_t) \, dW_t, \, \, \, X_0 = 1 \, \text {a.s.}$$
Assume $\sigma$ is regular enough that the above SDE admits a globally defined solution.
Suppose $|\sigma(x)| \to 0$ as $x \to 0$.
Question: Is it true that almost surely, $X_t > 0$ for all $t$?
It seems like the Dambis-Dubins-Schwarz theorem may help, but I’m not sure how to turn it into a proof.