Stopped Sigma-Algebra of Pointwise Minimum of Stopping Times
Theorem
Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a discrete-time 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:
- $\FF_{S \wedge T} = \FF_S \cap \FF_T$
where $\FF_{\paren \cdot}$ denotes the stopped $\sigma$-algebra.
Proof
From the definition of pointwise minimum, we have:
- $S \wedge T \le S$
and:
- $S \wedge T \le T$
Then from Stopped Sigma-Algebra preserves Inequality between Stopping Times, we have:
- $\FF_{S \wedge T} \subseteq \FF_S$
and:
- $\FF_{S \wedge T} \subseteq \FF_T$
so that:
- $\FF_{S \wedge T} \subseteq \FF_S \cap \FF_T$
Now let $A \in \FF_S \cap \FF_T$
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:
- $\set {\omega \in \Omega : \map {\paren {S \wedge T} } \omega \le t} = \set {\omega \in \Omega : \map S \omega \le t} \cup \set {\omega \in \Omega : \map T \omega \le t}$
Then, from Intersection Distributes over Union:
- $A \cap \set {\omega \in \Omega : \map {\paren {S \wedge T} } \omega \le t} = \paren {A \cap \set {\omega \in \Omega : \map S \omega \le t} } \cup \paren {A \cap \set {\omega \in \Omega : \map T \omega \le t} }$
Since $A \in \FF_S$, we have:
- $A \cap \set {\omega \in \Omega : \map S \omega \le t} \in \FF_t$
Since $A \in \FF_T$, we have:
- $A \cap \set {\omega \in \Omega : \map T \omega \le t} \in \FF_t$
So, since $\FF_t$ is closed under finite union, we have:
- $\paren {A \cap \set {\omega \in \Omega : \map S \omega \le t} } \cup \paren {A \cap \set {\omega \in \Omega : \map T \omega \le t} } \in \FF_t$
so:
- $A \cap \set {\omega \in \Omega : \map {\paren {S \wedge T} } \omega \le t} \in \FF_t$
$\blacksquare$