First Hitting Problems for Markov Chains That Converge to a Geometric Brownian Motion. (11th October 2011) - Lefebvre Lefebvre, Mario Mario; Kounta Kounta, Moussa 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 probability p j that the chain, starting from 1 + j Δ x, will hit N before 1, as well as the expected number d j of transitions needed to end the game. In the limit when Δ x and the time Δ t between the transitions decrease to zero appropriately, the Markov chain tends to a geometric Brownian motion. We show that p j and d j Δ t tend to the corresponding quantities for the geometric Brownian motion.