Least Time at which Discrete-Time Adapted Stochastic Process equals or exceeds Real Number 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 $\sequence {X_n}_{n \ge 0}$ be a discrete-time $\sequence {\FF_n}_{n \ge 0}$-adapted stochastic process.
Let $\lambda \in \R$.
Let:
- $T = \inf \set {n \in \Z_{\ge 0} : X_n \ge \lambda}$
Then $T$ is a stopping time with respect to $\sequence {\FF_n}_{n \ge 0}$.
Proof
Note that for $t \in \Z_{\ge 0}$ and $\omega \in \Omega$, we have:
- $\inf \set {n \in \Z_{\ge 0} : \map {X_n} \omega \ge \lambda} \le t$
- $\map {X_s} \omega \ge \lambda$ for some $s \le t$.
That is, we have:
- $\ds \set {\omega \in \Omega : \inf \set {n \in \Z_{\ge 0} : \map {X_n} \omega \ge \lambda} \le t} = \bigcup_{s \in \Z_{\ge 0}, \, 0 \le s \le t} \set {\omega \in \Omega : \map {X_s} \omega \ge \lambda}$
Since $\sequence {X_n}_{n \ge 0}$ is $\sequence {\FF_n}_{n \ge 0}$-adapted, we have:
- $\set {\omega \in \Omega : \map {X_s} \omega \ge \lambda} \in \FF_s$ for each $0 \le s \le t$.
Since $\sequence {\FF_n}_{n \ge 0}$ is a filtration, we have:
- $\FF_s \subseteq \FF_t$
So:
- $\set {\omega \in \Omega : \map {X_s} \omega \ge \lambda} \in \FF_t$ for each $s \in \Z_{\ge 0}$ with $0 \le s \le t$.
So since $\FF_t$ is closed under finite union, we conclude:
- $\set {\omega \in \Omega : \inf \set {n \in \Z_{\ge 0} : \map {X_n} \omega \ge \lambda} \le t} \in \FF_t$
for each $t \in \Z_{\ge 0}$.
So $T$ is a stopping time with respect to $\sequence {\FF_n}_{n \ge 0}$.
$\blacksquare$