First Hitting Place Probabilities for a Discrete Version of the Ornstein-Uhlenbeck Process

Abstract

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 hitsNbefore 0 is computed explicitly. Similarly, the probability that the process hitsNbefore−Mis computed in the case when the state space is{−M,…,0,…,N}and the transition probabilitiespi,i+1are not necessarily the same wheniis positive andiis negative.