On ruin probability and aggregate claim representations for Pareto claim size distributions

Albrecher, Hansjörg; Kortschak, Dominik

doi:10.1016/j.insmatheco.2009.08.005

Cited by 10 publications

(14 citation statements)

References 19 publications

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“…Note that the direct evaluation of 1 −L f (s) for |s| small is numerically not stable (column TWS14), so we use the simple identity 1 −L f (s) = e s (1 − αs α Γ(−α, s)) + (1 − e s ) and evaluate (1 − αs α Γ(−α, s)) with the series of Formula 6.5.29 of Abramowitz & Stegun [2] (truncated after sufficiently many summands, cf. also Albrecher & Kortschak [3]). For the term (1 − e s ) we use a series expansion as well.…”

Section: Us-pareto Distributionmentioning

confidence: 89%

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On the efficient evaluation of ruin probabilities for completely monotone claim distributions

Albrecher

Avram²,

Kortschak

2010

Journal of Computational and Applied Mathematics

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In this paper we propose a highly accurate approximation procedure for ruin probabilities in the classical collective risk model, which is based on a quadrature/rational approximation procedure proposed by Trefethen et al. [12]. For a certain class of claim size distributions (which contains the completely monotone distributions) we give a theoretical justification for the method. We also show that under weaker assumptions on the claim size distribution, the method may still perform reasonably well in some cases. This in particular provides an efficient alternative to a related method proposed by Thorin [10]. A number of numerical illustrations for the performance of this procedure is provided for both completely monotone and other types of random variables. − 1 2π − e δu+ιxuL h (δ + ιx) dx. Note now thatL h (s) =L h (s), where w is the complex conjugate of w. Indeed, for Re(s) > 0 we have thatL h (s) = ∞ 0 e −st h(t) dt, from which the observation follows. For Re(s) ≤ 0 we may always find an s 0 such that Re(s 0 ) > 0 and |s − s 0 | < |s 0 |, from which

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Section: Us-pareto Distributionmentioning

confidence: 89%

“…For the case of evaluating ψ(u), we get that at leastL ψ (s) → 0 for s → ∞ andL ψ (s) is meromorphic with infinitely many poles in D, whose residuals go to zero exponentially fast as u → ∞, cf. Albrecher & Kortschak [3]. Also, the limit at the negative real axis exists.…”

Section: Pareto Distributionmentioning

confidence: 93%

On the efficient evaluation of ruin probabilities for completely monotone claim distributions

Albrecher

Avram²,

Kortschak

2010

Journal of Computational and Applied Mathematics

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“…This has turned out to be difficult. Practical computation algorithms have been developed for claim sizes with different versions of the pure Pareto distribution; see [17,18] and [1]. Our approach for calculating the ruin probability will apply to a wide range of levels of u and to a wide variety of claim size distributions.…”

Section: Introductionmentioning

confidence: 99%

Calculation of ruin probabilities for a dense class of heavy tailed distributions

Bladt¹,

Nielsen

Samorodnitsky

2014

Scandinavian Actuarial Journal

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Abstract. In this paper we propose a class of infinite-dimensional phase-type distributions with finitely many parameters as models for heavy tailed distributions. The class of finite-dimensional distributions is dense in the class of distributions on the positive reals and may hence approximate any such distribution. We prove that formulas from renewal theory, and with a particular attention to ruin probabilities, which are true for common phase-type distributions also hold true for the infinite-dimensional case. We provide algorithms for calculating functionals of interest such as the renewal density and the ruin probability. It might be of interest to approximate a given heavy-tailed distribution of some other type by a distribution from the class of infinite-dimensional phase-type distributions and to this end we provide a calibration procedure which works for the approximation of distributions with a slowly varying tail. An example from risk theory, comparing ruin probabilities for a classical risk process with Pareto distributed claim sizes, is presented and exact known ruin probabilities for the Pareto case are compared to the ones obtained by approximating by an infinite-dimensional hyper-exponential distribution.

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“…This can be done by complex analysis techniques in an analogous way as in Albrecher and Kortschak (2009) and one arrives at the mixing density…”

Section: Example 25 (Dependent Gamma Claims)mentioning

confidence: 99%

“…Then (4) applies and one still has the freedom to choose the mixing distribution of Θ to identify a number of formulas for different dependence structures and marginals. However, this is only of limited usefulness, as for Pareto distributed claims there is no fully explicit formula for ψ θ (u) available (but see Ramsay, 2003 andKortschak, 2009 for integral representations of ψ θ (u)).…”

Section: Example 25 (Dependent Gamma Claims)mentioning

confidence: 99%

Explicit ruin formulas for models with dependence among risks

Albrecher

Constantinescu

Loisel

2011

Insurance: Mathematics and Economics

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We show that a simple mixing idea allows to establish a number of explicit formulas for ruin probabilities and related quantities in collective risk models with dependence among claim sizes and among claim inter-occurrence times. Examples include compound Poisson risk models with completely monotone marginal claim size distributions that are dependent according to Archimedean survival copulas as well as renewal risk models with dependent inter-occurrence times.

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On ruin probability and aggregate claim representations for Pareto claim size distributions

Cited by 10 publications

References 19 publications

On the efficient evaluation of ruin probabilities for completely monotone claim distributions

On the efficient evaluation of ruin probabilities for completely monotone claim distributions

Calculation of ruin probabilities for a dense class of heavy tailed distributions

Explicit ruin formulas for models with dependence among risks

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