Pointwise Minimum of Stopping Times is Stopping Time
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Theorem
Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a filtered probability space.
Let $T$ and $S$ be stopping times with respect to $\sequence {\FF_n}_{n \ge 0}$.
Let $S \wedge T$ be the pointwise minimum of $S$ and $T$.
Then $S \wedge T$ is a stopping time with respect to $\sequence {\FF_n}_{n \ge 0}$.
Proof
We have, for $t \in \Z_{\ge 0}$ and $\omega \in \Omega$:
- $\map {\paren {S \wedge T} } \omega \le t$ if and only if $\map S \omega \le t$ or $\map T \omega \le t$
That is:
- $\set {\omega \in \Omega : \map {\paren {S \wedge T} } \omega \le t} = \set {\omega \in \Omega : \map S \omega \le t} \cup \set {\omega \in \Omega : \map T \omega \le t}$
Since $S$ and $T$ are stopping times with respect to $\sequence {\FF_n}_{n \ge 0}$, we have:
- $\set {\omega \in \Omega : \map S \omega \le t} \in \FF_t$
and:
- $\set {\omega \in \Omega : \map T \omega \le t} \in \FF_t$
So, since $\FF_t$ is closed under finite union, we have:
- $\set {\omega \in \Omega : \map {\paren {S \wedge T} } \omega \le t} \in \FF_t$
$\blacksquare$