We study a renewal risk model in which the surplus process of the insurance company is modeled by a compound fractional Poisson process. We establish the long-range dependence property of this non-stationary process. Some results for the ruin probabilities are presented in various assumptions on the distribution of the claim sizes.AMS 2000 subject classifications: Primary 60G22, 60G55, 91B30; secondary 60K05, 33E12.
We study a renewal risk model in which the surplus process of the insurance company is modeled by a compound fractional Poisson process. We establish the long-range dependence property of this non-stationary process. Some results for the ruin probabilities are presented in various assumptions on the distribution of the claim sizes.AMS 2000 subject classifications: Primary 60G22, 60G55, 91B30; secondary 60K05, 33E12.
A risk process with constant premium rate c and Poisson arrivals of claims is considered. A threshold r is defined for claim interarrival times, such that if k consecutive interarrival times are larger than r, then the next claim has distribution G. Otherwise, the claim size distribution is F . Asymptotic expressions for the infinite horizon ruin probabilities are given for both light-and the heavy-tailed cases. A basic observation is that the process regenerates at each G-claim. Also an approach via Markov additive processes is outlined, and heuristics are given for the distribution of the time to ruin.
In the compound Poisson risk model, several strong hypotheses may be found too restrictive to describe accurately the evolution of the reserves of an insurance company. This is especially true for a company that faces natural disaster risks like earthquake or flooding. For such risks, claim amounts are often inter-dependent and they may also depend on the history of the natural phenomenon. The present paper is concerned with a situation of this kind, where each claim amount depends on the previous claim inter-arrival time, or on past claim inter-arrival times in a more complex way. Our main purpose is to evaluate, for large initial reserves, the asymptotic finite-time ruin probabilities of the company when the claim sizes have a heavy-tailed distribution. The approach is based more particularly on the analysis of spacings in a conditioned Poisson process.
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