My question is :
Show that if $\tau'$ is another hitting time for the filtration $\mathcal{F}_n$, then
$$\tau := \inf\{n\geq\tau' : X_n \in B, \mbox{where $B$ is a Borel set}\}$$
is also a hitting time.
Does someone know how to prove this ?
Thank you.
Hint: $$\{\tau \leq k\} = \{\tau' \leq \tau \leq k\} = \bigcup_{j=0}^k (\{\tau'=j\} \cap \{j \leq \tau \leq k\})$$
and so
$$\{\tau \leq k\} = \bigcup_{j=0}^k \left( \{\tau'=j\} \cap\bigcup_{i=j}^k \{X_i \in B\} \right) \in \mathcal{F}_k$$