First hitting place probabilities for a discrete version of the Ornstein-Uhlenbeck Process

A Markov chain with state space {0, ..., N} and transition probabilities depending on the current state is studied. The chain can be considered as a discrete Ornstein-Uhlenbeck process. The probability that the process hits N before 0 is computed explicitly. Similarly, the probability that the process hits N before −M is computed in the case when the state space is {−M, ..., 0, ..., N} and the transition probabilities pi,i1 are not necessarily the same when i is positive and i is negative.