Let (X, S) = {(Xn, Sn); n ≧ 0} be a Markov random walk with finite state space. For a ≦ 0 < b define the stopping times τ = inf{n:Sn > b} and T = inf{n:Sn ∉ (a, b)}. The diffusion approximations of a one-barrier probability P{τ < ∞ | X0 = i}, and a two-barrier probability P{ST ≧ b | X0 = i} with correction terms are derived. Furthermore, to approximate the above ruin probabilities, the limiting distributions of overshoot for a driftless Markov random walk are involved.
