First hitting problems for Markov chains that converge to a geometric Brownian motion

We consider a discrete-time Markov chain with state space {1,1+∆x,...,1+k∆x = N}. We compute explicitly the probability pj that the chain, starting from 1 + j∆x, will hit N before 1, as well as the expected number dj of transitions needed to end the game. In the limit when ∆x and the time ∆t between the transitions decrease to zero appropriately, the Markov chain tends to a geometric Brownian motion. We show that pj and dj∆t tend to the corresponding quantities for the geometric Brownian motion.