Stopped process mathematics , a stopped process is a stochastic process that is forced to assume the same value after a prescribed time.Definition Let stopped process is defined for and byExamples Gambling Consider a gambler playing roulette . X t 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. 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 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 Y , and, since the gambler cannot continue to play after he/she has exhausted his/her resources, X is the stopped process Y τ 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 :
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