Stochastic 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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Examples

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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}.

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Brownian motion

Let $B:[0,+\infty )\times \Omega \to \mathbb {R}$ be a 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 $t\geq \tau (\omega )$.