Ruin probabilities for a regenerative Poisson gap generated risk process

Abstract

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.

On Exceedance Times for Some Processes with Dependent Increments

“…As demonstrated by the examples in [8] (and later papers, of which Asmussen & Biard [4] is a recent instance), this result covers a large number of examples. Foss & Zachary [19] gave a similar result in the case of a modulated random walk.…”

“…. , τ rw − 1 as well as some bias in their distribution (expected to be small as well); this was realized in [4], with the consequent that some result there are heuristic. To overcome this difficulty, we present an approach to results of type Theorem 1.2 which is novel and combines the ideas from [5] and a sample-path analysis developed in [10,17,19].…”

“…For the tail asymptotics of ξ, the −R term may often be neglected (see [4] for some preliminary discussion and [1] for a more complete picture). Also, with light-tailed claims one may frequently approximate X 1 + · · · + X M RΛ by mRΛ; the relevant large deviations arguments are given in detail in [8] and will not be repeated here.…”

On the appropriate choice of the risk model

“…In this model [3] the point process of claim amount arrival instants is a homogeneous Poisson process with global rate λ. A threshold d is introduced for claim interarrival times, such that if the two immediately preceding consecutive both newly observed interarrival times are larger than d the claim size distribution has density f 2 in all other cases the density function is f 1 .…”

“…For instance it has been incorporated into the stochastic model that the distributions of the times between consecutive claim occurrences times may depend on the last previous claim amount [1] or that claim sizes depend on the past of the point process of instants when claims are presented [2,3]. Results on ruin probabilities and related quantities have been published in several papers under such assumptions.…”