Stopped process

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

Let

Then the stopped process ${\displaystyle X^{\tau }}$ is defined for ${\displaystyle t\geq 0}$ and ${\displaystyle \omega \in \Omega }$ by

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).

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.

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