International Journal of Mathematics and Mathematical Sciences
Volume 2009 (2009), Article ID 909835, 12 pages
Research Article

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

Département de Mathématiques et de Génie Industriel, École Polytechnique, C.P. 6079, Succursale Centre-ville, Montréal, QC, H3C 3A7, Canada

Received 2 August 2009; Accepted 10 November 2009

Academic Editor: Gideon Schechtman

Copyright © 2009 Mario Lefebvre and Jean-Luc Guilbault. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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,i+1 are not necessarily the same when i is positive and i is negative.