Let be a stochastic process adopted to the filtration of sub--fields of . A random variable whose values lie in is a stopping time for the filtration if the event for every finite .

Remarks

• 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.
• The -field at time , or , intuitively, is just a collection of events, or, “information” on all possible outcomes that are available up to, and including, the th bet.
• The inclusion of in the range of is to add a case where there is no stopping time; the process continues into infinity.
• The stopping time can be alternatively characterized as a non-negative integer-valued random variable, such that the event it is independent of .

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 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 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 of plays that it takes the gambler to stop is a stopping time.