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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.

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