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[1]) 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).
Let
\tau
(\Omega,lF,(lFn)n,P)
N\cup\{+infty\}
\tau
F=((lFn)n
\{\tau=n\}\inlFn
n
n
n
Let
\tau
(\Omega,lF,(lFt)t,P)
T
T=[0,+infty)
\tau
F=(lFt)t
\{\tau\leqt\}\inlFt
t\inT
Let
\tau
(\Omega,lF,(lFt)t,P)
T
\tau
X=(Xt)t
Xt:=\begin{cases}1&ift<\tau\ 0&ift\geq\tau\end{cases}
is adapted to the filtration
F=(lFt)t
Some authors explicitly exclude cases where
\tau
+infty
\tau
T
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:
To illustrate the more general definition of stopping time, consider Brownian motion, which is a stochastic process
(Bt)t\geq
Bt
(\Omega,l{F},P)
l{F}t
| -1 | |
| (B | |
| s) |
(A)
0\leqs\leqt
A\subseteqR
l{F}t
\tau:=t0
t0
a\inR.
\tau:=inf\{t\geq0\midBt=a\}
\tau:=inf\{t\geq1\midBs>0foralls\in[t-1,t]\}
\left(\Omega,l{F},\left\{l{F}t\right\}t,P\right)
\tau1\wedge\tau2
\tau1\vee\tau2
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,[2] 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 τ is a stopping time, then Xτ is used to denote the process X stopped at time τ.
| \tau | |
| X | |
| t=X |
min(t,\tau)
| 1 | |
| \{\taun>0\ |
| 1 | |
| \{\taun>0\ |
Locally integrable process. A non-negative and increasing process X is locally integrable if there exists a sequence of stopping times τn increasing to infinity, such that}X^ \right ]<\infty for each n.\operatorname{E}\left
[1 \{\taun>0\
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''<sub>''n''</sub> satisfying ''τ''<sub>''n''</sub> < ''τ'' whenever ''τ'' > 0. The sequence ''τ''<sub>''n''</sub> is said to ''announce'' ''τ'', and predictable stopping times are sometimes known as ''announceable''. Examples of predictable stopping times are [[hitting time]]s of continuous and 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 τn, where τn 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 = τn for some n) = 1, where τn 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 processes.
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 ∞.
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.