First Hitting Problems for Markov Chains That Converge to a Geometric Brownian MotionReport as inadecuate




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ISRN Discrete MathematicsVolume 2011 2011, Article ID 346503, 15 pages

Research Article

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

Département de Mathématiques et de Statistique, Université de Montréal, C.P. 6128, Succursale Centre-Ville, Montréal, QC, H3C 3J7, Canada

Received 1 July 2011; Accepted 21 July 2011

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

Copyright © 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+Δ𝑥,…,1+𝑘Δ𝑥=𝑁}. We compute explicitly the probability 𝑝𝑗 that the chain, starting from 1+𝑗Δ𝑥, will hit N before 1, as well as the expected number 𝑑𝑗 of transitions needed to end the game. In the limit when Δ𝑥 and the time Δ𝑡 between the transitions decrease to zero appropriately, the Markov chain tends to a geometric Brownian motion. We show that 𝑝𝑗 and 𝑑𝑗Δ𝑡 tend to the corresponding quantities for the geometric Brownian motion.





Author: Mario Lefebvre and Moussa Kounta

Source: https://www.hindawi.com/



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