Stable Lévy motion approximation in collective risk theory

Furrer, Hansjörg; Michna, Zbigniew; Weron, Aleksander

doi:10.1016/s0167-6687(97)00008-5

Cited by 37 publications

(32 citation statements)

References 11 publications

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“…The results we present here extend previous work of Furrer [6], Albrecher et al [1], Tsai and Wilmott [15] and Kolkovska and Martín-González [11]. Weak approximations in risk theory have been used in Sarkar and Sen [13] in the case α = 2 and λ 2 = 0, and in Furrer et al [7] in the case λ 2 = 0 to estimate ruin probabilities within finite time horizon.…”

Section: Introductionsupporting

confidence: 82%

“…Using the weak approximation for the α-stable process given in Furrer et al [7], we construct a sequence of two-sided classical risk processes which weakly approximates the process V α in the Skorokhod space, and prove the convergence of the corresponding expected discounted penalty functions. Afterward we obtain the Laplace transform of ϕ and a defective renewal equation for ϕ.…”

Section: Introductionmentioning

confidence: 99%

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Path functionals of a class of Lévy insurance risk processes

Kolkovska¹,

Martín-González²

2016

COSA

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Abstract. We study expected discounted penalty functionals for a class of Lévy processes having a component given by the difference of two independent Poisson compound processes, and a perturbation term given by an α-stable process. We obtain a formula for the Laplace transforms of the expected discounted penalty functionals, as well as explicit representations of such functionals as infinite series of convolutions of given functions. We illustrate our results in some particular examples.

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Section: Introductionsupporting

confidence: 82%

Section: Introductionmentioning

confidence: 99%

Path functionals of a class of Lévy insurance risk processes

Kolkovska¹,

Martín-González²

2016

COSA

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“…We now turn to showing that in general, the ALPs we consider do converge to a stable Lévy motion, and we moreover apply the results of Furrer et al (1997) and Michna (2005) to ALPs. For our sequence of iid losses {X k } ∞ k=1 satisfying Equation (2) and {N(nt)} ∞ n=1 , a sequence of renewal processes, if:…”

Section: Definition 4 (Billingsleymentioning

confidence: 95%

“…where Z α (t) is an α-stable Lévy motion (Furrer et al 1997). Observe that if L is a constant, say d, then we may rewrite Equation (5) as:…”

Section: Definition 4 (Billingsleymentioning

confidence: 99%

Stable Weak Approximation at Work in Index-Linked Catastrophe Bond Pricing

Burnecki

Giuricich

2017

Risks

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Abstract:We consider the subject of approximating tail probabilities in the general compound renewal process framework, where severity data are assumed to follow a heavy-tailed law (in that only the first moment is assumed to exist). By using the weak convergence of compound renewal processes to α-stable Lévy motion, we derive such weak approximations. Their applicability is then highlighted in the context of an existing, classical, index-linked catastrophe bond pricing model, and in doing so, we specialize these approximations to the case of a compound time-inhomogeneous Poisson process. We emphasize a unique feature of our approximation, in that it only demands finiteness of the first moment of the aggregate loss processes. Finally, a numerical illustration is presented. The behavior of our approximations is compared to both Monte Carlo simulations and first-order single risk loss process approximations and compares favorably.

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“…Lévy processes are commonly used in risk theory and the related literature has become considerable, see e.g. Furrer et al (1997), Yang and Zhang (2001), Klüppelberg et al (2004), Morales (2004), Avram et al (2007), Kyprianou and Palmowski (2007), Biffis andMorales (2010), etc. Palmowski andPistorius (2009) propose a limiting approximation for the finite time first passage probability of Lévy processes, with the same kind of asymptotics considered in this article.…”

Section: Introductionmentioning

confidence: 99%

Importance sampling approximations to various probabilities of ruin of spectrally negative Lévy risk processes

Gatto

2014

Applied Mathematics and Computation

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This article provides importance sampling algorithms for computing the probabilities of various types ruin of spectrally negative Lévy risk processes, which are ruin over the infinite time horizon, ruin within a finite time horizon and ruin past a finite time horizon. For the special case of the compound Poisson process perturbed by diffusion, algorithms for computing probabilities of ruins by creeping (i.e. induced by the diffusion term) and by jumping (i.e. by a claim amount) are provided. It is shown that these algorithms have either bounded relative error or logarithmic efficiency, as t, x → ∞, where t > 0 is the time horizon and x > 0 is the starting point of the risk process, with y = t/x held constant and assumed either below or above a certain constant. Key words and phrasesBounded relative error; exponential tilt; Legendre-Fenchel transform; logarithmic efficiency; Lundberg conjugated measure; ruin due to creeping and to jump; ruin past a finite time horizon, within a finite and the infinite time horizons; saddlepoint approximation; short and long time horizons.

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Stable Lévy motion approximation in collective risk theory

Cited by 37 publications

References 11 publications

Path functionals of a class of Lévy insurance risk processes

Path functionals of a class of Lévy insurance risk processes

Stable Weak Approximation at Work in Index-Linked Catastrophe Bond Pricing

Importance sampling approximations to various probabilities of ruin of spectrally negative Lévy risk processes

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