In probability theory, in particular in the study of stochastic processes, a stopping time (also Markov time) is a specific type of “random time”.
The theory of stopping rules and stopping times can be analysed in probability and statistics, notably in the optional stopping theorem. Also, stopping times are frequently applied in mathematical proofs- to “tame the continuum of time”, as Chung put it in his book (1982). Stopping times are hitting times.
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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, ..., Xt. 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.
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:
Hitting times can be important examples of stopping times. However, while it is relatively straightforward to show that all stopping times are hitting times (see Fischer (2011)), 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.
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 is a stopping time, then X is used to denote the process X stopped at time .
Then, X is said to locally satisfy some property P if there exists a sequence of stopping times n, which increases to infinity and for which the processes satisfy property P. Common examples, with time index set I = [0,∞), are as follows;
Stopping times, with time index set I = [0,∞), are often divided into one of several types depending on whether it is possible to predict when they are about to occur.
A stopping time is predictable if it is equal to the limit of an increasing sequence of stopping times n satisfying n < whenever > 0. The sequence n is said to announce , and predictable stopping times are sometimes known as announceable. Examples of predictable stopping times are hitting times of continuous and adapted processes. If is the first time at which a continuous and real valued process X is equal to some value a, then it is announced by the sequence n, where n is the first time at which X is within a distance of 1/n of a.
Accessible stopping times are those that can be covered by a sequence of predictable times. That is, stopping time is accessible if, P(=n for some n) = 1, where n are predictable times.
A stopping time is totally inaccessible if it can never be announced by an increasing sequence of stopping times. Equivalently, P( = σ < ∞) = 0 for every predictable time σ. Examples of totally inaccessible stopping times include the jump times of Poisson processes.
Every stopping time can be uniquely decomposed into an accessible and totally inaccessible time. That is, there exists a unique accessible stopping time σ and totally inaccessible time υ such that = σ whenever σ < ∞, = υ whenever υ < ∞, and = ∞ whenever σ = υ = ∞. Note that in the statement of this decomposition result, stopping times do not have to be almost surely finite, and can equal ∞.