As the title suggests, I am not sure how to show if $\tau=\inf\{t:X_t=\sup_{0\leq t\leq T}X_t\}$ is a stopping time or not, where $\{X_t\}$ is an adapted process. Intuitively, I see that it is not a stopping time from the finance context — if $X_t$ refers to stock price and we want to exercise an American call option, then if $\tau$ was a stopping time this would contradict the assertion that early exercise of an American call option is not optimal. However, this is as far I can go, as my intuition tells me this is not in $\mathcal{F}_t$ generated by $X_t$, but how should I start proving this rigorously, and a bigger concern is what about $\mathcal{F}_{t^+}$? Would be nice to see in particular for the geometric Brownian motion case too.
(NB: I am asking this in the Math SE instead of QF SE as I am searching for a more mathematical justification, but I am placing my American option argument in here for some context.)
Assume, for the sake of contradiction, that $\tau$ is a stopping time.
I will assume $X$ Brownian motion so that it's clearer.
Since $0 \leq \tau \leq T$, we can you the optional stopping time theorem and say that $\mathbb{E}[X_\tau] = \mathbb{E}[X_0] = 0$.
However, it's clear that $X_\tau > 0$ almost surely, so we obtain a contradiction.