On the probability of ruin in a Markov-modulated risk model

In this paper, we consider a Markov-modulated risk model in which the claim inter-arrivals, claim sizes and premiums are influenced by an external Markovian environment process. A system of Laplace transforms of non-ruin probabilities, given the initial environment state, is established from a system of integro-differential equations derived by Reinhard [Reinhard, J.M., 1984. On a class of semi-Markov risk models obtained as classical risk models in a Markovian environment. ASTIN Bull., 14, 23–43]. In the two-state model, explicit formulas for non-ruin probabilities are obtained when the initial reserve is zero or when both claim size distributions belong to the K n -family, n ∈ N + . Examples are given with claim sizes that have exponential, Erlang and a mixture of exponential distributions.