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In
probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
, in particular in the study of
stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...
es, a stopping time (also Markov time, Markov moment, optional stopping time or optional time ) is a specific type of “random time”: a
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
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 In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty, but it has probability 0. ...
lead to a decision to stop at some finite time. Stopping times occur in
decision theory Decision theory (or the theory of choice; not to be confused with choice theory) is a branch of applied probability theory concerned with the theory of making decisions based on assigning probabilities to various factors and assigning numerical ...
, 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 \tau be a random variable, which is defined on the
filtered probability space Filtration is a physical separation process that separates solid matter and fluid from a mixture using a ''filter medium'' that has a complex structure through which only the fluid can pass. Solid particles that cannot pass through the filter m ...
(\Omega, \mathcal F, (\mathcal F_n)_, P) with values in \mathbb N \cup \. Then \tau is called a stopping time (with respect to the
filtration Filtration is a physical separation process that separates solid matter and fluid from a mixture using a ''filter medium'' that has a complex structure through which only the fluid can pass. Solid particles that cannot pass through the filter ...
\mathbb F= ((\mathcal F_n)_ ), if the following condition holds: : \ \in \mathcal F_n for all n Intuitively, this condition means that the "decision" of whether to stop at time n must be based only on the information present at time n, not on any future information.


General case

Let \tau be a random variable, which is defined on the
filtered probability space Filtration is a physical separation process that separates solid matter and fluid from a mixture using a ''filter medium'' that has a complex structure through which only the fluid can pass. Solid particles that cannot pass through the filter m ...
(\Omega, \mathcal F, (\mathcal F_t)_, P) with values in T. In most cases, T=
filtration_ Filtration_is_a_physical_separation_process_that_separates_solid_matter_and__fluid_from_a_mixture_using_a_''filter_medium''_that_has_a_complex_structure_through_which_only_the_fluid_can_pass._Solid_particles_that_cannot_pass_through_the_filter__...
__\mathbb_F=_(\mathcal_F_t)__),_if_the_following_condition_holds: :_\_\in_\mathcal_F_t__for_all__t_\in_T_


__As_adapted_process_

Let__\tau__be_a_random_variable,_which_is_defined_on_the_filtered_probability_space_ Filtration_is_a_physical_separation_process_that_separates_solid_matter_and_fluid_from_a_mixture_using_a_''filter_medium''_that_has_a_complex_structure_through_which_only_the_fluid_can_pass._Solid_particles_that_cannot_pass_through_the_filter_m_...
__(\Omega,_\mathcal_F,_(\mathcal_F_t)_,_P)__with_values_in__T._Then__\tau__is_called_a_stopping_time_iff_the_stochastic_process_ In_probability_theory_and_related_fields,_a_stochastic_()_or_random_process_is_a_mathematical_object_usually_defined_as_a_family_of_random_variables._Stochastic_processes_are_widely_used_as_mathematical_models_of_systems_and_phenomena_that_appea_...
__X=(X_t)_,_defined_by :_X_t:=_\begin_1_&_\text_t_<_\tau_\\_0_&\text_t_\geq_\tau_\end_ is_adapted_process.html" "title=",+ \infty) . Then \tau is called a stopping time (with respect to the
filtration Filtration is a physical separation process that separates solid matter and fluid from a mixture using a ''filter medium'' that has a complex structure through which only the fluid can pass. Solid particles that cannot pass through the filter ...
\mathbb F= (\mathcal F_t)_ ), if the following condition holds: : \ \in \mathcal F_t for all t \in T


As adapted process

Let \tau be a random variable, which is defined on the
filtered probability space Filtration is a physical separation process that separates solid matter and fluid from a mixture using a ''filter medium'' that has a complex structure through which only the fluid can pass. Solid particles that cannot pass through the filter m ...
(\Omega, \mathcal F, (\mathcal F_t)_, P) with values in T. Then \tau is called a stopping time iff the
stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...
X=(X_t)_, defined by : X_t:= \begin 1 & \text t < \tau \\ 0 &\text t \geq \tau \end is adapted process">adapted In biology, adaptation has three related meanings. Firstly, it is the dynamic evolutionary process of natural selection that fits organisms to their environment, enhancing their evolutionary fitness. Secondly, it is a state reached by the po ...
 to the filtration \mathbb F= (\mathcal F_t)_


Comments

Some authors explicitly exclude cases where \tau can be + \infty , whereas other authors allow \tau to take any value in the closure of T.


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 are 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 Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position insi ...
, which is a stochastic process (B_t)_, where each B_t is a random variable defined on the probability space (\Omega, \mathcal, \mathbb). We define a filtration on this probability space by letting \mathcal_t be the ''σ''-algebra generated by all the sets of the form (B_s)^(A) where 0\leq s \leq t and A\subseteq \mathbb is a
Borel set In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are named ...
. Intuitively, an event ''E'' is in \mathcal_t 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 \tau:=t_0 is (trivially) a stopping time; it corresponds to the stopping rule "stop at time t_0". * Let a\in\mathbb. Then \tau:=\inf \ 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 \tau:=\inf \. 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 \left(\Omega, \mathcal, \left\_, \mathbb\right) then their minimum \tau _1 \wedge \tau _2, their maximum \tau _1 \vee \tau _2, and their sum ''τ''1 + ''τ''2 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 time In the study of stochastic processes in mathematics, a hitting time (or first hit time) is the first time at which a given process "hits" a given subset of the state space. Exit times and return times are also examples of hitting times. Definitions ...
s 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 ''X''''τ'' is used to denote the process ''X'' stopped at time ''τ''. : X^\tau_t=X_ Then, ''X'' is said to locally satisfy some property ''P'' if there exists a sequence of stopping times τ''n'', which increases to infinity and for which the processes :\mathbf_X^ satisfy property ''P''. Common examples, with time index set ''I'' = , ∞), are as follows:
Local martingale process. A process ''X'' is a local martingale if it is càdlàg and there exists a sequence of stopping times τ''n'' increasing to infinity, such that :\mathbf_X^ is a martingale (probability theory), martingale for each ''n''.
Locally integrable process. A non-negative and increasing process ''X'' is locally integrable if there exists a sequence of stopping times ''τ''''n'' increasing to infinity, such that :\operatorname \left mathbf_X^ \right \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 ''τ'' is predictable if it is equal to the limit of an increasing sequence of stopping times ''τ''''n'' satisfying ''τ''''n'' < ''τ'' whenever ''τ'' > 0. The sequence ''τ''''n'' is said to ''announce'' ''τ'', and predictable stopping times are sometimes known as ''announceable''. Examples of predictable stopping times are hitting times of continuous and Adapted process, adapted processes. If ''τ'' 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 ''τ''''n'', where ''τ''''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 ''τ'' is accessible if, P(''τ'' = ''τ''''n'' for some ''n'') = 1, where ''τ''''n'' are predictable times. A stopping time ''τ'' is totally inaccessible if it can never be announced by an increasing sequence of stopping times. Equivalently, P(''τ'' = ''σ'' < ∞) = 0 for every predictable time ''σ''. Examples of totally inaccessible stopping times include the jump times of
Poisson process In probability, statistics and related fields, a Poisson point process is a type of random mathematical object that consists of points randomly located on a mathematical space with the essential feature that the points occur independently of one ...
es. Every stopping time ''τ'' 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 ''τ'' = ''σ'' whenever ''σ'' < ∞, ''τ'' = ''υ'' whenever ''υ'' < ∞, and ''τ'' = ∞ 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 ∞.


Stopping rules in clinical trials

Clinical trials in medicine often perform interim analysis, in order to determine whether the trial has already met its endpoints. However, interim analysis create the risk of false-positive results, and therefore stopping boundaries are used to determine the number and timing of interim analysis (also known as alpha-spending, to denote the rate of false positives). At each of R interim tests, the trial is stopped if the likelihood is below a threshold p, which depends on the method used. See
Sequential analysis In statistics, sequential analysis or sequential hypothesis testing is statistical analysis where the sample size is not fixed in advance. Instead data are evaluated as they are collected, and further sampling is stopped in accordance with a pre- ...
.


See also

*
Optimal stopping In mathematics, the theory of optimal stopping or early stopping : (For French translation, secover storyin the July issue of ''Pour la Science'' (2009).) is concerned with the problem of choosing a time to take a particular action, in order to ...
*
Odds algorithm The odds algorithm (or Bruss algorithm) is a mathematical method for computing optimal strategies for a class of problems that belong to the domain of optimal stopping problems. Their solution follows from the ''odds strategy'', and the importance ...
*
Secretary problem The secretary problem demonstrates a scenario involving optimal stopping theory For French translation, secover storyin the July issue of ''Pour la Science'' (2009). that is studied extensively in the fields of applied probability, statistics, an ...
*
Hitting time In the study of stochastic processes in mathematics, a hitting time (or first hit time) is the first time at which a given process "hits" a given subset of the state space. Exit times and return times are also examples of hitting times. Definitions ...
*
Stopped process In mathematics, a stopped process is a stochastic process that is forced to assume the same value after a prescribed (possibly random) time. Definition Let * (\Omega, \mathcal, \mathbb) be a probability space; * (\mathbb, \mathcal) be a measurable ...
*
Disorder problem In the study of stochastic processes in mathematics, a disorder problem or quickest detection problem (formulated by Kolmogorov) is the problem of using ongoing observations of a stochastic process to detect as soon as possible when the probabilisti ...
* Début theorem *
Sequential analysis In statistics, sequential analysis or sequential hypothesis testing is statistical analysis where the sample size is not fixed in advance. Instead data are evaluated as they are collected, and further sampling is stopped in accordance with a pre- ...


References


Further reading

* Thomas S. Ferguson
“Who solved the secretary problem?”, Stat. Sci. vol. 4, 282–296, (1989).

An introduction to stopping times.
*
F. Thomas Bruss Franz Thomas Bruss is Emeritus Professor of Mathematics at the Université Libre de Bruxelles, where he had been director of "Mathématiques Générales" and co-director of the probability chair, and where he continues his research as invited profe ...
, “Sum the odds to one and stop”, Annals of Probability, Vol. 4, 1384–1391,(2000) * * * * {{DEFAULTSORT:Stopping Time Stochastic processes Optimal decisions