Large claims approximations for risk processes in a Markovian environment

Let 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 and the claim size distribution is when the environment is in state . Assuming that there is a subset of E for which the satisfy, as that i.e. , for some probability distribution H whose tail is a subexponential density, and for the remaining , it is shown that for some explicit constant . 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.