In mathematics, a stopped process is a stochastic process that is forced to assume the same value after a prescribed (possibly random) time.
Let
Then the stopped process is defined for and by
Consider a gambler playing roulette. Xt 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 Yt denote what the gambler's holdings would be if he/she could obtain unlimited credit (so Y can attain negative values).
Let be one-dimensional standard Brownian motion starting at zero.