Table of Contents
ISRN Discrete Mathematics
Volume 2011 (2011), Article ID 346503, 15 pages
http://dx.doi.org/10.5402/2011/346503
Research Article

First Hitting Problems for Markov Chains That Converge to a Geometric Brownian Motion

1Département de Mathématiques et de Génie Industriel, École Polytechnique de Montréal, C.P. 6079, Succursale Centre-Ville, Montréal, QC, Canada H3C 3A7
2Département de Mathématiques et de Statistique, Université de Montréal, C.P. 6128, Succursale Centre-Ville, Montréal, QC, Canada H3C 3J7

Received 1 July 2011; Accepted 21 July 2011

Academic Editors: C.-K. Lin and B. Zhou

Copyright Β© 2011 Mario Lefebvre and Moussa Kounta. 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.

Abstract

We consider a discrete-time Markov chain with state space {1,1+Ξ”π‘₯,…,1+π‘˜Ξ”π‘₯=𝑁}. We compute explicitly the probability 𝑝𝑗 that the chain, starting from 1+𝑗Δπ‘₯, will hit N before 1, as well as the expected number 𝑑𝑗 of transitions needed to end the game. In the limit when Ξ”π‘₯ and the time Δ𝑑 between the transitions decrease to zero appropriately, the Markov chain tends to a geometric Brownian motion. We show that 𝑝𝑗 and 𝑑𝑗Δ𝑑 tend to the corresponding quantities for the geometric Brownian motion.