{{Short description|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 * <math>(\Omega, \mathcal{F}, \mathbb{P})</math> be a probability space; * <math>(\mathbb{X}, \mathcal{A})</math> be a 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 with respect to some filtration <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 playing 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 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. 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 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 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
==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