Large claims approximations for risk processes in a Markovian environment

Let [psi]i(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 [beta]j and the claim size distribution is Bj when the environment is in state j[set membership, variant]E. Assuming that there is a subset of E for which the Bj satisfy, as x --> [infinity] that 1 - Bj(x) [approximate] bj(1 - H(x)); i.e. (1 - Bj(x))/(1 - H(x))-->bj [set membership, variant] (0, [infinity]), 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 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.