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.

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Definition

Let

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. Xt 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 Yt denote what the gambler's holdings would be if he/she could obtain unlimited credit (so Y can attain negative values).

\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 Yτ.

Brownian motion

Let B�: [0, %2B \infty) \times \Omega \to \mathbb{R} be a one-dimensional standard Brownian motion starting at zero.

\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