# stopping time

Let $(\mathcal{F}_{t})_{t\in\mathbb{T}}$ be a filtration  (http://planetmath.org/FiltrationOfSigmaAlgebras) on a set $\Omega$. A random variable  $\tau$ taking values in $\mathbb{T}\cup\{\infty\}$ is a stopping time for the filtration $(\mathcal{F}_{t})$ if the event $\{\tau\leq t\}\in\mathcal{F}_{t}$ for every $t\in\mathbb{T}$.

Remarks

Examples. A gambler has $1,000 and plays the slot machine at$1 per play.

1. 1.

The gambler stops playing when his capital is depleted. The number $\tau=n_{1}$ of plays that it takes the gambler to stop is a stopping time.

2. 2.

The gambler stops playing when his capital reaches $2,000. The number $\tau=n_{2}$ of plays that it takes the gambler to stop is a stopping time. 3. 3. The gambler stops playing when his capital either reaches$2,000, or is depleted, which ever comes first. The number $\tau=\operatorname{min}(n_{1},n_{2})$ of plays that it takes the gambler to stop is a stopping time.

Title stopping time StoppingTime 2013-03-22 14:41:13 2013-03-22 14:41:13 gel (22282) gel (22282) 11 gel (22282) Definition msc 60K05 msc 60G40 DoobsOptionalSamplingTheorem PredictableStoppingTime