ON APPROXIMATIONS OF RISK PROCESS WITH RENEWAL ARRIVALSIN-STABLE DOMAIN
Z. Michna
Abstract: In this paper we approximate risk process by an -stable Lévy motion
(). We consider two conditions imposed on the value of the premium rate. The
first one assumes that the premiums exceed only slightly the expected claims (heavy traffic)
and the second one assumes that the premiums are much larger than the average claims (light
traffic). We consider the distribution of claim sizes belonging to the domain of attraction of an
-stable law and the process counting claims is a renewal process constructed from random
variables belonging to the domain of attraction of an -stable law. Comparing
and we obtain three different asymptotic risk processes. In the classical
model We get a Brownian diffusion approximation which fits first two moments. If
we get Mittag-Leffler distribution for the infinite time ruin probability, and if
we obtain exponential distribution for the infinite time ruin probability.