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 + Δ x , … , 1 + k Δ x = N } . We compute explicitly the probabilityp jthat the chain, starting from 1 + j Δ x , will hit N before 1, as well as the expected numberd jof 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 thatp jandd j Δ t tend to the corresponding quantities for the geometric Brownian motion.