Closed operations

If $T_1$ and $T_2$ are stopping times, then:

  • $\min(T_1,T_2)$ is a stopping time, because
$$ \{\min(T_1,T_2)\le n\}=\{T_1\le n\}\cup\{T_2\le n\}. $$
  • $\max(T_1,T_2)$ is a stopping time, because
$$ \{\max(T_1,T_2)\le n\}=\{T_1\le n\}\cap\{T_2\le n\}. $$
  • In discrete time, $T_1+T_2$ is a stopping time, because

$$ {T_1+T_2\le n}

\bigcup_{k=0}^n {T_1=k}\cap{T_2\le n-k}. $$

Non-closed operations

In general, the following are not stopping times:

  • $T_1-T_2$
  • $T_1T_2$

The issue is that these expressions can depend on a random reference time, so deciding the event at absolute time $n$ may require future information.

Useful consequence

The second smallest of three stopping times is still a stopping time, because

$$ T_{(2)}

\max\bigl(\min(T_1,T_2),\min(T_1,T_3),\min(T_2,T_3)\bigr). $$