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

Then the stopped process is defined for and by

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).

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 be a one-dimensional standard Brownian motion starting at zero.

Then the stopped Brownian motion will evolve as per usual up until the random time , and will thereafter be constant with value : i.e., for all .

See also

References