I have encountered the following question for homework, with our lecturer only requiring us to have a basic idea about stopping times. The question is as follows:
Let $X(t)$ be an Ito process and suppose that $a\neq b$ and:
$\tau_{a} = inf\{ t>0: X(t) = a \}$
and
$\tau_{b} = inf\{ t>0: X(t) = b \}$
Is $\tau_{a} \wedge \tau_{b}$ a stopping time. Explain your answer?
I understand that $\tau_{a}$ and $\tau_{b}$ are both first passage times, however considering them in conjunction, my reasoning has stalled.
Any feedback would be greatly appreciated!