Characterization of Stopping Times with respect to Right-Limit Filtration

== Theorem ==
Let $\struct {\Omega, \Sigma, \sequence {\FF_t}_{t \ge 0}, \Pr}$ be a continuous-time filtered probability space.
Let $T : \Omega \to \closedint 0 \infty$ be a random variable.
Let $\sequence {\GG_t}_{t \ge 0}$ be the right-limit filtration associated with $\sequence {\FF_t}_{t \ge 0}$.
:$(1) \quad$ $T$ is a stopping time with respect to $\sequence {\GG_t}_{t \ge 0}$
:$(2) \quad$ for each $t \in \hointr 0 \infty$, we have $\set {\omega \in \Omega : \map T \omega < t} \in \FF_t$
:$(3) \quad$ for each $t \in \hointr 0 \infty$, the pointwise minimum $T \wedge t$ is $\FF_t$-measurable. 
== Proof ==
=== $(1)$ implies $(2)$ ===
Suppose $T$ is a stopping time with respect to $\sequence {\GG_t}_{t \ge 0}$ and let $t \in \hointr 0 \infty$.
Then, for each $n \in \N$ we have: 
:$\ds \set {\omega \in \Omega : \map T \omega \le t - \frac 1 n} \in \GG_{t - \frac 1 n}$
Since by the definition of the right-limit filtration, we have: 
:$\ds \GG_{t - \frac 1 n} = \bigcap_{s > t - \frac 1 n} \FF_s$
we have: 
:$\GG_{t - \frac 1 n} \subseteq \FF_s$ for all $\ds s > t - \frac 1 n$
and in particular:
:$\GG_{t - \frac 1 n} \subseteq \FF_t$ 
and hence:
:$\ds \set {\omega \in \Omega : \map T \omega \le t - \frac 1 n} \in \FF_t$
Note that: 
:$\ds \set {\omega \in \Omega : \map T \omega < t} = \bigcup_{n \mathop = 1}^\infty \set {\omega \in \Omega : \map T \omega \le t - \frac 1 n}$ 
Since $\FF_t$ is closed under countable union, we have: 
:$\ds \bigcup_{n \mathop = 1}^\infty \set {\omega \in \Omega : \map T \omega \le t - \frac 1 n} \in \FF_t$
and hence:
:$\set {\omega \in \Omega : \map T \omega < t} \in \FF_t$
=== $(2)$ implies $(1)$ ===
Suppose that for each $t \in \hointr 0 \infty$, we have $\set {\omega \in \Omega : \map T \omega < t} \in \FF_t$
Let $t \in \hointr 0 \infty$ and $s > t$. 
We have: 
:$\ds \set {\omega \in \Omega : \map T \omega < t + \frac 1 n} \in \FF_{t + \frac 1 n}$
Since: 
:$\ds t + \frac 1 n \to t$ 
and the sequence is decreasing, there exists $N_t \in \N$ such that: 
:$\ds t + \frac 1 n < s$ 
for $n \ge N_t$. 
Then, for $n \ge N_t$, we have: 
:$\ds \set {\omega \in \Omega : \map T \omega < t + \frac 1 n} \in \FF_s$ 
since $\sequence {\FF_t}_{t \ge 0}$ is a filtration. 
Now note that: 
:$\ds \set {\omega \in \Omega : \map T \omega \le t} = \bigcap_{n \mathop = N_t}^\infty \set {\omega \in \Omega : \map T \omega < t + \frac 1 n}$
Since $\FF_s$ is closed under countable intersection, we have: 
:$\ds \set {\omega \in \Omega : \map T \omega \le t} \in \FF_s$ for each $s > t$. 
So, by the definition of the right-limit filtration, we have: 
:$\ds \set {\omega \in \Omega : \map T \omega \le t} \in \GG_t$
So $T$ is a stopping time with respect to $\sequence {\GG_t}_{t \ge 0}$.
=== $(1)$ implies $(3)$ ===
Suppose that $T$ is a stopping time with respect to $\sequence {\GG_t}_{t \ge 0}$.
Then for each $s < t$, we have: 
:$\set {\omega \in \Omega : \map T \omega \le s} \in \GG_s$ 
and hence:
:$\set {\omega \in \Omega : \map T \omega \le s} \in \FF_t$
since $\GG_s \subseteq \FF_t$ for $s < t$.
Now note that if $\map T \omega \le s$ for $s < t$, we have: 
:$\min \set {\map T \omega, t} \le s$ 
Conversely, if: 
:$\min \set {\map T \omega, t} \le s < t$ 
we must have $\map T \omega \le s$. 
So, we have: 
:$\set {\omega \in \Omega : \map {\paren {T \wedge t} } \omega \le s} \in \FF_t$
If $s \ge t$, then: 
:$\set {\omega \in \Omega : \map {\paren {T \wedge t} } \omega \le s} = \Omega \in \FF_t$
Since $\FF_t$ is a $\sigma$-algebra.
We conclude that $T \wedge t$ is $\FF_t$-measurable.
=== $(3)$ implies $(2)$ ===
Suppose that for each $t \in \hointr 0 \infty$, the pointwise minimum $T \wedge t$ is $\FF_t$-measurable.
Fix $t \in \hointr 0 \infty$, we want to show that: 
:$\set {\omega \in \Omega : \map T \omega < t} \in \FF_t$ 
We have:
:$\set {\omega \in \Omega : \map {\paren {T \wedge t} } \omega \le s} \in \FF_t$
for all $s \in \hointr 0 \infty$.
That is: 
:$\set {\omega \in \Omega : \map T \omega \le s} \in \FF_t$ 
for $s < t$, so that: 
:$\ds \set {\omega \in \Omega : \map T \omega \le t - \frac 1 n} \in \FF_t$
for each $n \in \N$.
Since $\FF_t$ is closed under countable union, we have:
:$\ds \bigcup_{n \mathop = 1}^\infty \set {\omega \in \Omega : \map T \omega \le t - \frac 1 n} \in \FF_t$ 
That is: 
:$\set {\omega \in \Omega : \map T \omega < t} \in \FF_t$
== Sources ==
* : Proposition $3.6$
Category:Stopping Times
Category:Right-Limit Filtrations of Sigma-Algebras