Large claims approximations for risk processes in a Markovian environment

AbstractLet ψi(u) be the probability of ruin for a risk process which has initial reserve u and evolves in a finite Markovian environment E with initial state i. Then the arrival intensity is βj and the claim size distribution is Bj when the environment is in state j∈E. Assuming that there is a subset of E for which the Bj satisfy, as x → ∞ that 1 − Bj(x) ≈ bj(1 − H(x)); i.e. (1 − Bj(x))(1 − H(x))→bj ∈ (0, ∞), for some probability distribution H whose tail is subexponential density, and 1 − Bj(x) = o(1 − H(x)) for the remaining Bj, it is shown that ψi(u) ≈ ci ∫u∞ (1 − H(x)) dx for some explicit constant ci. By time-reversion, similar results hold for the tail of the waiting time in a Markov-modulated M/G/1 queue whenever the service times satisfy similar conditions