In probability theory, in particular in the study of stochastic processes, a stopping time (also Markov time, Markov moment, optional stopping time or optional time) is a specific type of “random time”: a random variable whose value is interpreted as the time at which a given stochastic process exhibits a certain behavior of interest. A stopping time is often defined by a stopping rule, 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 finite time.
Stopping times occur in decision theory, and the optional stopping theorem is an important result in this context. Stopping times are also frequently applied in mathematical proofs to “tame the continuum of time”, as Chung put it in his book (1982).
Definition
Discrete time
Let be a random variable, which is defined on the filtered probability space with values in . Then is called a stopping time (with respect to the filtration ), if the following condition holds: for all Intuitively, this condition means that the "decision" of whether to stop at time must be based only on the information present at time , not on any future information.
General case
Let be a random variable, which is defined on the filtered probability space with values in . In most cases, . Then is called a stopping time (with respect to the filtration ), if the following condition holds: for all
As adapted process
Let be a random variable, which is defined on the filtered probability space with values in . Then is called a stopping time iff the stochastic process , defined by
is adapted to the filtration
Comments
Some authors explicitly exclude cases where can be , whereas other authors allow to take any value in the closure of .
Examples
To illustrate some examples of random times that are stopping rules and some that are not, consider a gambler playing roulette with a typical house edge, starting with $100 and betting $1 on red in each game:
Playing exactly five games corresponds to the stopping time τ = 5, and is a stopping rule. Playing until they either run out of money or have played 500 games is a stopping rule. Playing until they obtain the maximum amount ahead they will ever be is not a stopping 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 used to denote the process X stopped at time τ.