First Hitting Problems for Markov Chains That Converge to a Geometric Brownian Motion. (11th October 2011)

• Record Type: Journal Article
• Authors: Lefebvre, Mario; Kounta, Moussa
• Other Names: Lin C.-K. Academic Editor.; Zhou B. Academic Editor.
• Is Part Of: ISRN discrete mathematics. Volume 2011(2011)
• Publication Date: 2011-10-11
• Subjects: Discrete mathematics -- Periodicals; Computer science -- Mathematics; Computer science -- Mathematics; Periodicals; 511.1
• DOI: 10.5402/2011/346503
• Languages: English
• ISSNs: 2090-7788

Abstract

Abstract : We consider a discrete-time Markov chain with state space{ 1, 1 + Δ x, …, 1 + k Δ x = N } . We compute explicitly the probabilityp j that the chain, starting from1 + j Δ x, will hit N before 1, as well as the expected numberd j of transitions needed to end the game. In the limit whenΔ x and the timeΔ t between the transitions decrease to zero appropriately, the Markov chain tends to a geometric Brownian motion. We show thatp j andd j Δ t tend to the corresponding quantities for the geometric Brownian motion.