# Stopped process

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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

- be a probability space;
- be a measurable space;
- be a stochastic process;
- be a stopping time with respect to some filtration of .

Then the **stopped process** is defined for and by

## 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

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

- Stopping at a deterministic time : if , then the stopped Brownian motion will evolve as per usual up until time , and thereafter will stay constant: i.e., for all .

- Stopping at a random time: define a random stopping time by the first hitting time for the region :

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 .