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:

$A \cap \set {\omega \in \Omega : \map {\paren {S \wedge T} } \omega \le t} \in \FF_t$

$\blacksquare$