Let $X_t$ be a Ito diffusion satisfying the SDE $dX_t=\mu(X_t)dt+\sigma(X_t)dW_t$ with $\mu$ and $\sigma$ being Lipschitz and $X_0=0$. Assume that $\sigma(x)>0$. Can we prove that $\tau=0$ almost surely where $\tau:=\inf\{t\geq0:X_t>0\}$? If not, what extra assumption needs to be made for this to be true. I know at least this works for Brownian motion.
Is there any reference for such property?
1. By Girsanov's theorem, the laws of $X$ and of the solution $Y$ of $dY_t=\sigma(Y_t)dW_t$ are locally absolutely continuous, so you can assume without loss of generality that $\mu=0$.
2. When $\mu=0$ your process $X$ is a local martingale, hence a time change of Brownian motion—for which the conclusion is known to be true. How is your question affected by time change?