S. Asmussen et al., LARGE CLAIMS APPROXIMATIONS FOR RISK PROCESSES IN A MARKOVIAN ENVIRONMENT, Stochastic processes and their applications, 54(1), 1994, pp. 29-43
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 elem
ent of E. Assuming that there is a subset of E for which the B-j satis
fy, 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 s
ome 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 t
he tail of the waiting time in a Markov-modulated M/G/1 queue whenever
the service times satisfy similar conditions.