Stopping 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).
be a random variable, which is defined on the filtered probability space
is called a stopping time (with respect to the filtration
must be based only on the information present at time
be a random variable, which is defined on the filtered probability space
is called a stopping time (with respect to the filtration
be a random variable, which is defined on the filtered probability space
is called a stopping time if the stochastic process
Some authors explicitly exclude cases where
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
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
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.
Then, X is said to locally satisfy some property P if there exists a sequence of stopping times τn, which increases to infinity and for which the processes satisfy property P. Common examples, with time index set I = [0, ∞), are as follows: Local martingale process.
A process X is a local martingale if it is càdlàg[clarification needed] and there exists a sequence of stopping times τn increasing to infinity, such that
A non-negative and increasing process X is locally integrable if there exists a sequence of stopping times τn increasing to infinity, such that
A stopping time τ is predictable if it is equal to the limit of an increasing sequence of stopping times τn satisfying τn < τ whenever τ > 0.
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.
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.
Examples of totally inaccessible stopping times include the jump times of Poisson processes.
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.
