At time 0, I have $2. At times 1, 2, . . . , I play a game in which I bet $1. With probability p, I win the game, and with probability 1  p, I lose the game. My goal is to increase my capital to $4, and as soon as I do, the game is over. The game is also over if my capital is reduced to $0. If we define Xt to be my capital position after the time t game (if any) is played, then X0, X1, . . . , Xt may be viewed as a discrete-time stochastic process.

Note that X0  2 is a known constant, but X1 and later Xt’s are random. For example, with probability p, X1  3, and with probability 1  p, X1  1. Note that if Xt  4, then Xt1 and all later Xt’s will also equal 4. Similarly, if Xt  0, then Xt1 and all later Xt’s will also equal 0. For obvious reasons, this type of situation is called a gambler’s ruin problem.