Stopped process

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title: "Stopped process" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["stochastic-processes"] description: "Stochastic process" topic_path: "general/stochastic-processes" source: "https://en.wikipedia.org/wiki/Stopped_process" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Stochastic 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{F}, \mathbb{P}) be a probability space;
• (\mathbb{X}, \mathcal{A}) be a measurable space;
• X : [0, + \infty) \times \Omega \to \mathbb{X} be a stochastic process;
• \tau : \Omega \to [0, + \infty] be a stopping time with respect to some filtration { \mathcal{F}_{t} | t \geq 0 } of {}\mathcal{F}.

Then the stopped process X^{\tau} is defined for t \geq 0 and \omega \in \Omega by

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

Examples

Gambling

Consider a gambler playing roulette. X**t denotes the gambler's total holdings in the casino at time t ≥ 0, which may or may not be allowed to be negative, depending on whether or not the casino offers credit. Let Y**t denote what the gambler's holdings would be if he/she could obtain unlimited credit (so Y can attain negative values).

• Stopping at a deterministic time: suppose that the casino is prepared to lend the gambler unlimited credit, and that the gambler resolves to leave the game at a predetermined time T, regardless of the state of play. Then X is really the stopped process Y**T, since the gambler's account remains in the same state after leaving the game as it was in at the moment that the gambler left the game.
• Stopping at a random time: suppose that the gambler has no other sources of revenue, and that the casino will not extend its customers credit. The gambler resolves to play until and unless he/she goes broke. Then the random time \tau (\omega) := \inf { t \geq 0 | Y_{t} (\omega) = 0 } is a stopping time for Y, and, since the gambler cannot continue to play after he/she has exhausted his/her resources, X is the stopped process Y**τ.

Brownian motion

Let B : [0, + \infty) \times \Omega \to \mathbb{R} be one-dimensional standard Brownian motion starting at zero.

• Stopping at a deterministic time T 0: if \tau (\omega) \equiv T, then the stopped Brownian motion B^{\tau} will evolve as per usual up until time T, and thereafter will stay constant: i.e., B_{t}^{\tau} (\omega) \equiv B_{T} (\omega) for all t \geq T.
• Stopping at a random time: define a random stopping time \tau by the first hitting time for the region { x \in \mathbb{R} | x \geq a }: \tau (\omega) := \inf { t 0 | B_{t} (\omega) \geq a }. Then the stopped Brownian motion B^{\tau} will evolve as per usual up until the random time \tau, and will thereafter be constant with value a: i.e., B_{t}^{\tau} (\omega) \equiv a for all t \geq \tau (\omega).

References

::callout[type=info title="Wikipedia Source"] This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page. ::