The density of the time to ruin for a Sparre Andersen process with Erlang arrivals and exponential claims

We derive a closed-form (infinite series) representation for the distribution of the ruin time for the Sparre Andersen model with exponentially distributed claims. This extends a recent result of Dickson et al. [Dickson, D.C.M., Hughes, B.D., Zhang, L., 2005.The density of the time to ruin for a Sparre Andersen process with Erlang arrivals and exponential claims. Scand. Actuar. J., 358-376] for such processes with Erlang inter-claim times. The derivation is based on transforming the original boundary crossing problem to an equivalent one on linear lower boundary crossing by a spectrally positive Lévy process. We illustrate our result in the cases of gamma, mixed exponential and inverse Gaussian inter-claim time distributions.

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“…Ruin probabilistic results for the special case F ∼ Erlang (2, λ) in the Sparre Andersen model have been obtained in [5, 6, 8-10, 25, 31] and in [24] and [14,15], in the case when claim inter-arrival times have distribution F ∼ Erlang(n, λ). In the latter case, [11] derive expressions for the density of the time to ruin in the special case of independent identically exponentially distributed claim amounts. Some research has also been performed beyond the Sparre Andersen assumption of independence of the times between consecutive claim arrivals.…”

Section: Introductionmentioning

confidence: 99%

Finite time non-ruin probability for Erlang claim inter-arrivals and continuous inter-dependent claim amounts

Ignatov¹,

Kaishev

2011

Stochastics

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This is the accepted version of the paper.This version of the publication may differ from the final published version. A closed form expression, in terms of some functions which we call exponential Appell polynomials, for the probability of non-ruin of an insurance company, in a finite-time interval is derived, assuming independent, non-identically Erlang distributed claim inter-arrival times, Permanent repository link. ., any continuous joint distribution of the claim amounts and any non-negative, non-decreasing real function, representing its premium income. In the special case when τ i ∼ Erlang (g i , λ) , i = 1, 2, . . . it is shown that our main result yields a formula for the probability of non-ruin expressed in terms of the classical Appell polynomials. We give another special case of our non-ruin probability formula for. ., i.e., when the inter-arrival times are non-identically exponentially distributed and also show that it coincides with the formula for Poisson claim arrivals, given in [18], when τ i ∼ Erlang(1, λ), i = 1, 2, . . .. The main result is extended further to a risk model in which inter-arrival times are dependent random variables, obtained by randomizing the Erlang shape or/and rate parameters. We give also some useful auxiliary results which characterize and express explicitly (and recurrently) the exponential Appell polynomials which appear in our finite time non-ruin probability formulae.

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The density of the time to ruin for a Sparre Andersen process with Erlang arrivals and exponential claims

Cited by 29 publications

References 6 publications

On the discounted penalty function in the renewal risk model with general interclaim times

On the discounted penalty function in the renewal risk model with general interclaim times

On the ruin time distribution for a Sparre Andersen process with exponential claim sizes

Finite time non-ruin probability for Erlang claim inter-arrivals and continuous inter-dependent claim amounts

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