Ruin problems in a discrete Markov risk model

In this paper, we extend the compound binomial risk model to a Markov dependent model in which the claim occurrence and the claim amount are both regulated by a discrete time Markov process. The explicit expression for the "discounted" joint probability function of the surplus before ruin and the deficit at ruin is derived when the initial surplus u=0, and a recursive formula to calculate such "discounted" joint probability function when the initial surplus u>0 is also obtained.