On the gambler’s ruin problem for a finite Markov chain

In this paper we present closed-form formulas for the solutions of the gambler’s ruin problem for a finite Markov chain where probabilities of winning or losing a particular game depending on the amount of the current fortune, from probability boundary conditions’ viewpoint, and provide some very simple closed forms which immediately lead to exact and explicit formulas for some special cases, depending on the relationships between the transition probabilities and the boundary condition probabilities.