Least Time at which Discrete-Time Adapted Stochastic Process equals or exceeds Real Number is Stopping Time

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:

$\inf \set {n \in \Z_{\ge 0} : \map {X_n} \omega \ge \lambda} \le t$

if and only if:

$\map {X_s} \omega \ge \lambda$ 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 $\sequence {\FF_n}_{n \ge 0}$.

$\blacksquare$