Stopped process

In mathematics, a stopped process is a stochastic process that is forced to assume the same value after a prescribed (possibly random) time.

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) . ==See also== Killed process ==References==

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