First Hitting Problems for Markov Chains That Converge to a Geometric Brownian Motion. (11th October 2011) - Lefebvre, Mario; Kounta, Moussa
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.