Stopped Sigma-Algebra of Constant Stopping Time coincides with Filtration

Theorem

Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a filtered probability space.

Let $\sequence {X_n}_{n \ge 0}$ be an $\sequence {\FF_n}_{n \ge 0}$-adapted stochastic process.

Let $n \in \Z_{\ge 0}$.

Let $T$ be a stopping time such that:

$\map T \omega = n$

for each $\omega \in \Omega$.

Let $\FF_T$ be the stopped $\sigma$-algebra associated with $T$.

Then:

$\FF_T = \FF_n$

Proof

We first show $\FF_n \subseteq \FF_T$.

Let $A \in \FF_n$.

Note that for $t \in \Z_{\ge 0}$, we have:

$\set {\omega \in \Omega : \map T \omega \le t} = \O$

if $t < n$.

That is:

$A \cap \set {\omega \in \Omega : \map T \omega \le t} = \O$

if $t < n$.

So certainly:

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

if $t < n$.

If $t \ge n$ we have:

$\set {\omega \in \Omega : \map T \omega \le t} = \Omega$

so that:

$A \cap \set {\omega \in \Omega : \map T \omega \le t} = A \in \FF_n$

So:

$A \cap \set {\omega \in \Omega : \map T \omega \le t} \in \FF_n$

for $t \ge n$.

Since $\sequence {\FF_n}_{n \ge 0}$ is a filtration and $t \ge n$, we have:

$\FF_n \subseteq \FF_t$

and so:

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

for $t \ge n$ as well.

So:

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

for all $t \in \Z_{\ge 0}$.

So $A \in \FF_T$.

So $\FF_n \subseteq \FF_T$.

Now we show $\FF_T \subseteq \FF_n$.

Let $A \in \FF_T$.

Then we have:

$A \cap \set {\omega \in \Omega : \map T \omega \le n} \in \FF_n$

But we have:

$\set {\omega \in \Omega : \map T \omega \le n} = \Omega$

so:

$A \cap \set {\omega \in \Omega : \map T \omega \le n} = A$

giving $A \in \FF_n$.

So $\FF_T \subseteq \FF_n$.

We conclude $\FF_T = \FF_n$.

$\blacksquare$