Stopping time

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).

Definition

A stopping time with respect to a sequence of random variables X₁, X₂, ... is a random variable $\tau$ with the property that for each t, the occurrence or non-occurrence of the event $\tau$ = t depends only on the values of X₁, X₂, ..., X_t. In some cases, the definition specifies that Pr( $\tau$ < ∞) = 1, or that $\tau$ 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 $(I,\leq )$ be an ordered index set (often $I=[0,\infty )$ or a compact subset thereof), and let $(\Omega ,{\mathcal {F}},\{{\mathcal {F}}_{t}\}_{t\in I},\mathbb {P} )$ be a filtered probability space, i.e. a probability space equipped with a filtration. Then a random variable $\tau :\Omega \to I$ is called a stopping time if $\{\tau \leq t\}\in {\mathcal {F}}_{t}$ for all $t$ in $I$ . Often, to avoid confusion, we call it a ${\mathcal {F}}_{t}$ -stopping time and explicitly specify the filtration. Speaking concretely, for $\tau$ to be a stopping time, it should be possible to decide whether or not $\{\tau \leq t\}$ has occurred on the basis of the knowledge of ${\mathcal {F}}_{t}$ , i.e., event $\{\tau \leq t\}$ is ${\mathcal {F}}_{t}$ -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.

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:

Playing one, and only one, game corresponds to the stopping time $\tau$ = 1, and is a stopping rule.
Playing until he either runs out of money or has played 500 games is a stopping rule.
Playing until he is the maximum amount ahead he 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 he doubles his money (borrowing if necessary if he goes into debt) is not a stopping rule, as there is a positive probability that he will never double his money. (Here it is assumed that there are limits that prevent the employment of a martingale system, or a variant thereof, such as each bet being triple the size of the last. Such limits could include betting limits but not limits to borrowing.)
Playing until he either doubles his money or runs out of money is a stopping rule, even though there is potentially no limit to the number of games he plays, since the probability that he stops in a finite time is 1.

Hitting times can be important examples of stopping times. However, while it is relatively straightforward to show that essentially all stopping times are hitting times (Fischer, 2013), 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 $\tau$ is a stopping time, then X^$\tau$ is used to denote the process X stopped at time $\tau$ .

X_{t}^{\tau }=X_{\min(t,\tau )}

Then, X is said to locally satisfy some property P if there exists a sequence of stopping times $\tau$ _n, which increases to infinity and for which the processes $1_{\{\tau _{n}>0\}}X^{\tau _{n}}$ satisfy property P. Common examples, with time index set I = [0,∞), are as follows;

(Local martingale) A process X is a local martingale if it is càdlàg and there exists a sequence of stopping times $\tau$ _n increasing to infinity, such that $1_{\{\tau _{n}>0\}}X^{\tau _{n}}$ is a martingale for each n.
(Locally integrable) A non-negative and increasing process X is locally integrable if there exists a sequence of stopping times $\tau$ _n increasing to infinity, such that $\mathbb {E} (1_{\{\tau _{n}>0\}}X^{\tau _{n}})<\infty$ for each n.

Types of stopping times

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 $\tau$ is predictable if it is equal to the limit of an increasing sequence of stopping times $\tau$ _n satisfying $\tau$ _n < $\tau$ whenever $\tau$ > 0. The sequence $\tau$ _n is said to announce $\tau$ , and predictable stopping times are sometimes known as announceable. Examples of predictable stopping times are hitting times of continuous and adapted processes. If $\tau$ 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 $\tau$ _n, where $\tau$ _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 $\tau$ is accessible if, P( $\tau$ = $\tau$ _n for some n) = 1, where $\tau$ _n are predictable times.

A stopping time $\tau$ is totally inaccessible if it can never be announced by an increasing sequence of stopping times. Equivalently, P( $\tau$ = σ < ∞) = 0 for every predictable time σ. Examples of totally inaccessible stopping times include the jump times of Poisson processes.

Every stopping time $\tau$ 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 $\tau$ = σ whenever σ < ∞, $\tau$ = υ whenever υ < ∞, and $\tau$ = ∞ 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 ∞.

If $\tau _{1}$ and $\tau _{2}$ are stopping times on the |filtered probability space, then $\tau _{1}\wedge \tau _{2},\tau _{1}\vee \tau _{2},\$and\$\tau _{1}+\tau _{2}$ are also stopping times.

References

Chung, Kai Lai (1982). Lectures from Markov processes to Brownian motion. Grundlehren der Mathematischen Wissenschaften No. 249. New York: Springer-Verlag. ISBN 0-387-90618-5.
Fischer, Tom (2013). "On simple representations of stopping times and stopping time sigma-algebras". Statistics and Probability Letters. 83 (1): 345–349. doi:10.1016/j.spl.2012.09.024.
Revuz, Daniel and Yor, Marc (1999). Continuous martingales and Brownian motion. Grundlehren der Mathematischen Wissenschaften No. 293 (Third edition ed.). Berlin: Springer-Verlag. ISBN 3-540-64325-7. {{cite book}}: |edition= has extra text (help)CS1 maint: multiple names: authors list (link)
H. Vincent Poor and Olympia Hadjiliadis (2008). Quickest Detection (First edition ed.). Cambridge: Cambridge University Press. ISBN 978-0-521-62104-5. {{cite book}}: |edition= has extra text (help)
Protter, Philip E. (2005). Stochastic integration and differential equations. Stochastic Modelling and Applied Probability No. 21 (Second edition (version 2.1, corrected third printing) ed.). Berlin: Springer-Verlag. ISBN 3-540-00313-4.