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

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.