Let $\{X_t, t \ge 0\}$ be a continuous stochastic process and adapted to the filtration $\{\mathcal{F}_t,t\ge 0 \}$ and consider
$$ \alpha = \inf\{t, |X_t|>1\}, $$ the first time the the process $X_t$ leaves the interval $[-1,1]$. Then can you help me to show that $\alpha$ is in fact stopping time?
The proof below is not correct. See an example by Did. The flaw in the proof: in fact we have $$ \{\alpha>t\} = \bigcup_n\{\alpha \geq t+\frac1n\} = \bigcup_{n}\bigcap_{s\in [0,t+\frac1n]}\{|X_s|\leq 1\} $$ and there appear events $\{|X_s|\leq 1\}$ for $s>t$.
By the definition, a random variable $\tau:\Omega\to [0,\infty]$ is a stopping time if and only if $$ \{\tau \leq t\}\in \mathscr F_t $$ for any $t\in [0,\infty)$. We have in your case $$ \{\alpha \leq t\} = \Omega\setminus \{\alpha > t\} = \Omega\setminus \bigcap_{s\in [0,t]}\{|X_s|\leq 1\} = \Omega\setminus\bigcap_{s\in \Bbb Q\cap [0,t]}\left\{|X_s|\leq 1\right\} \in \mathscr F_t $$ where we pass to the intersection over rational numbers only since $X$ has continuous trajectories.