I am trying to check if the process $S_t$ is a martingale, where $\mathrm dS_t = \frac{I_{S_t > 0}}{S_t} \mathrm dW_t$, $S_0 = 1$.
We know that $S_t$ is a local martingale because if we stop it at $\frac{1}{n}$ then the stopped process is an Ito Integral of a bounded process and thus is a martingale. So we also know $S_{\infty}$ exists since nonnegative local martingales are supermartingales and nonnegative supermartingales have a limit.
Now it seems intuitively clear that $S_t$ must hit zero on some positive measure set at some finite time and then stay there. Furthermore, when $S_t < M$, say, then $S_t$ is in some sense moving as fast at $\frac{1}{M}W_t$. There are many details missing here and I don't know how to fix them. Hopefully there is a simpler solution.
Any help would greatly be appreciated. Thank you.