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 B-j when the environment is in state j is an element of E. Assuming that there is a subset of E for which the B-j satisfy, as x --> infinity that 1 - B-j(x) similar to b(j)(1 - H(x)); i.e. (1 - B-j(x))/(1 - H(x)) --> b(j) is an element of (0, infinity), for some probability distribution H whose tail is a subexponential density, and 1 - B-j(x) = o(1 - H(x)) for the remaining B-j, it is shown that psi(i)(u) similar to c(i) integral(u)(infinity) (1 - H(x)) dx for some explicit constant c(i). 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.