Stopped process explained

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

(\Omega, l{F}, P ) be a probability space; ( X , l{A}) be a measurable space;X : [ 0 , + inf ty) x \Omega \to X be a stochastic process;\tau : \Omega \to [ 0 , + inf ty] be a stopping time with respect to some filtration \{ l{F} _{t} | t \geq 0 \} of {} l{F} .

Then the stopped process

X ^{\tau} is defined for t \geq 0 and \omega \in \Omega byX ^{\tau} _{t} (\omega) : = X _{min} } (\omega).

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

\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.

Brownian motion

Let

B : [ 0 , + inf ty) x \Omega \to 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 ^{\tau} _{t} (\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 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 ^{\tau} _{t} (\omega) \equiv a for all t \geq \tau (\omega) .

See also

• Killed process

References

• Robert G. Gallager. Stochastic Processes: Theory for Applications. Cambridge University Press, Dec 12, 2013 pg. 450