Stopping Time

A stopping time with respect to a sequence of random variables X₁, X₂, ... is a random variable with the property that for each t, the occurrence or non-occurrence of the event = t depends only on the values of X₁, X₂, ..., X_t. In some cases, the definition specifies that Pr( < ∞) = 1, or that be almost surely finite, although in other cases this requirement is omitted.

Another, more general definition may be given in terms of a filtration: Let be an ordered index set (often or a compact subset thereof), and let be a filtered probability space, i.e. a probability space equipped with a filtration. Then a random variable is called a stopping time if for all in . Often, to avoid confusion, we call it a -stopping time and explicitly specify the filtration. Speaking concretely, for to be a stopping time, it should be possible to decide whether or not has occurred on the basis of the knowledge of, i.e., event is -measurable.

Stopping times occur in decision theory, in which a stopping rule is characterized as a mechanism for deciding whether to continue or stop a process on the basis of the present position and past events, and which will almost always lead to a decision to stop at some time.