rule and does not provide a stopping time, as it requires information about the future as well as the present and past. Playing until they double their money (borrowing if necessary) is not a stopping rule, as there is a positive probability that they will never double their money. Playing until they either double their money or run out of money is a stopping rule, even though there is potentially no limit to the number of games they play, since the probability that they stop in a finite time is 1. To illustrate the more general definition of stopping time, consider Brownian motion, which is a stochastic process , where each is a random variable defined on the probability space . We define a filtration on this probability space by letting be the σ-algebra generated by all the sets of the form where and is a Borel set. Intuitively, an event E is in if and only if we can determine whether E is true or false just by observing the Brownian motion from time 0 to time t. Every constant is (trivially) a stopping time; it corresponds to the stopping rule "stop at time ". Let Then is a stopping time for Brownian motion, corresponding to the stopping rule: "stop as soon as the Brownian motion hits the value a." Another stopping time is given by . It corresponds to the stopping rule "stop as soon as the Brownian motion has been positive over a contiguous stretch of length 1 time unit." In general, if τ1 and τ2 are stopping times on then their minimum , their maximum , and their sum τ1 + τ2 are also stopping times. (This is not true for differences and products, because these may require "looking into the future" to determine when to stop.) Hitting times like the second example above can be important examples of stopping times. While it is relatively straightforward to show that essentially all stopping times are hitting times, it can be much more difficult to show that a certain hitting time is a stopping time. The latter types of results are known as the Début theorem. Localization Stopping times are frequently used to generalize certain properties of stochastic processes to situations in which the required property is satisfied in only a local sense. First, if X is a process and τ is a stopping time, then Xτ is