Suppose $(B_t)_{t\geq 0}$ is a Brownian motion and that $S_t = \exp (B_t-\frac{t}{2})$. By the martingale convergence theorem, $S_t\to S_\infty$, some random constant, a.s..
It seems that we should have $\mathbb{P}[S_\infty=0]=1$, but I'm not sure how to prove it. What's the best approach here?
Thank you.