martingale, stopping times

Question

Let $\sigma$ and $\tau$ be stopping times, prove that $(\sigma+\tau) \land T$ (T is the end-time point) is also a stopping time. So I have already proved that $\sigma+\tau$ is a stopping time, if I can prove that $T$ is also a stopping time then I would be finished? But is the end time automatically a stopping time? or do I have to use another approach with a case distinction? (e.g. $(\sigma+\tau) > T$ and $(\sigma+\tau) \leq T$)

Answer 1

It is immediate that $$ \{T\leqslant t\}\in \mathcal F_t $$ for all $t$. Now, since $\sigma+\tau$ and $T$ are stopping times, we have $$ \{(\sigma+\tau)\wedge T\leqslant t\} = \{\sigma+\tau\leqslant t\}\cup \{T\leqslant t\}\in\mathcal F_t $$ for all $t$, and hence $(\sigma+\tau)\wedge T$ is a stopping time.