stopping time

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

Remarks

• The set $𝕋$ is the index set for the time variable $t$, and the $\sigma$-algebra ${ℱ}_t$ 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 $\{\tau \le t\}$ is known at time $t$.

• In discrete time situations, where $𝕋=\{0,1,2,\mathrm{\dots }\}$, the condition that $\{\tau \le t\}\in {ℱ}_t$ is equivalent to requiring that $\{\tau =t\}\in {ℱ}_t$. This is not true for continuous time cases where $𝕋$ 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, $\{\tau \le t\}\in \sigma (X_s:s\le t)$.

• The first time that an adapted process $X_t$ hits a given value or set of values is a stopping time. The inclusion of $\mathrm{\infty }$ 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. 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. 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. The gambler stops playing when his capital either reaches $2,000, or is depleted, which ever comes first. The number $\tau =\min (n_1,n_2)$ of plays that it takes the gambler to stop is a stopping time.