I was trying to prove that the solution $X$ to the one dimensional SDE $dX_t = \sigma(X_t)dW_t$ (where $\sigma$ is a real valued Borel measurable function, $W$ is a 1d Brownian Motion) cannot explode, i.e. $S= \infty$ a.s., where $S= \inf\{t\geq 0: X_t \notin \mathbb{R}\}$. I was able to show that $\{\int^{t \min S} _0\sigma^2(X_s)ds= \infty\}$ has probability 0 for all $t\geq 0$ (using time changed brownian motion and law of iterated logarithm to arrive at a contradiction to almost sure path continuity of the solution in the extended real). How do I procede from here to conclude that $P\{S= \infty\}=1$? I was thinking of using Ito's isometry $E[\int^{t \min S} _0\sigma^2(X_s)ds]= E[(\int^{t \min S} _0\sigma(X_s)dW_s)^2]$ $\Rightarrow X_{t \min S} - X_0= \int^{t \min S} _0\sigma(X_s)dW_s \in \mathbb{R}$ a.s., but the expectation of the Ito integral could be unbounded.
This is an exercise from Karatzas & Shreve
Thanks!