# 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

• The set $\mathbb{T}$ is the index set for the time variable $t$, and the $\sigma$-algebra $\mathcal{F}_{t}$ is the collection of all events which are observable up to and including time $t$. Then, the condition that $\tau$ is a stopping time means that the outcome of the event $\{\tau\leq t\}$ is known at time $t$.

• In discrete time situations, where $\mathbb{T}=\{0,1,2,\ldots\}$, the condition that $\{\tau\leq t\}\in\mathcal{F}_{t}$ is equivalent to requiring that $\{\tau=t\}\in\mathcal{F}_{t}$. This is not true for continuous time cases where $\mathbb{T}$ is an interval of the real numbers and hence uncountable, due to the fact that $\sigma$-algebras are not in general closed under taking uncountable unions of events.

• A random time $\tau$ is a stopping time for a stochastic process $(X_{t})$ if it is a stopping time for the natural filtration of $X$. That is, $\{\tau\leq t\}\in\sigma(X_{s}:s\leq t)$.

• The first time that an adapted process $X_{t}$ hits a given value or set of values is a stopping time. The inclusion of $\infty$ into the range of $\tau$ is to cover the case where $X_{t}$ never hits the given values.

• Stopping time is often used in gambling, when a gambler stops the betting process when he reaches a certain goal. The time it takes to reach this goal is generally not a deterministic one. Rather, it is a random variable depending on the current result of the bet, as well as the combined information from all previous bets.

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