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.

2more comments