# Stopped process

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{{Short description|Stochastic process}}
In [mathematics](/source/mathematics), a '''stopped process''' is a [stochastic process](/source/stochastic_process) that is forced to assume the same value after a prescribed (possibly random) time.

==Definition==
Let
* <math>(\Omega, \mathcal{F}, \mathbb{P})</math> be a [probability space](/source/probability_space); 
* <math>(\mathbb{X}, \mathcal{A})</math> be a [measurable space](/source/measurable_space);
* <math>X : [0, + \infty) \times \Omega \to \mathbb{X}</math> be a stochastic process;
* <math>\tau : \Omega \to [0, + \infty]</math> be a [stopping time](/source/stopping_rule) with respect to some [filtration](/source/Filtration_(mathematics)) <math>\{ \mathcal{F}_{t} | t \geq 0 \}</math> of <math>{}\mathcal{F}</math>.

Then the '''stopped process''' <math>X^{\tau}</math> is defined for <math>t \geq 0</math> and <math>\omega \in \Omega</math> by

:<math>X_{t}^{\tau} (\omega) := X_{\min \{ t, \tau (\omega) \}} (\omega).</math>

==Examples==

===Gambling===

Consider a [gambler](/source/Gambling) playing [roulette](/source/roulette). ''X''<sub>''t''</sub> 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](/source/casino) offers credit. Let ''Y''<sub>''t''</sub> 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''<sup>''T''</sup>, 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](/source/Gambler's_ruin). Then the random time <math display=block>\tau (\omega) := \inf \{ t \geq 0 | Y_{t} (\omega) = 0 \}</math> 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''<sup>''τ''</sup>.

===Brownian motion===
Let <math>B : [0, + \infty) \times \Omega \to \mathbb{R}</math> be one-dimensional standard [Brownian motion](/source/Brownian_motion) starting at zero.

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

==See also==
* [Killed process](/source/Killed_process)

==References==
*{{cite book |first=Robert G. |last=Gallager |title=Stochastic Processes: Theory for Applications |publisher=Cambridge University Press |year=2013 |page=450 |isbn=978-1-107-03975-9 }}

{{DEFAULTSORT:Stopped Process}}
Category:Stochastic processes

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Adapted from the Wikipedia article [Stopped process](https://en.wikipedia.org/wiki/Stopped_process) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Stopped_process?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.