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Stopped process

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. X<sub>t</sub> 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<sub>t</sub> 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<sup>T</sup>, 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<sup>τ</sup>.

Brownian motion

Let be 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 .

See also

References