Exponential functional of a new family of Lévy processes and self-similar continuous state branching processes with immigration

Patie, Pierre

doi:10.1016/j.bulsci.2008.10.001

Cited by 52 publications

(68 citation statements)

References 31 publications

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“…Recent work however has produced many completely explicit examples of scale functions (e.g. : Hubalek and Kyprianou, 2008;Patie, 2008;Chaumont et al, 2009) and in particular Hubalek and Kyprianou (2008); Kyprianou and Rivero (2008) and Patie and Kyprianou (unpublished manuscript) outline a method from which many more examples can be computed than the articles themselves had the space for.…”

Section: Introductionmentioning

confidence: 98%

A note on scale functions and the time value of ruin for Lévy insurance risk processes

Biffis

Kyprianou

2010

Insurance: Mathematics and Economics

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a b s t r a c tWe examine discounted penalties at ruin for surplus dynamics driven by a general spectrally negative Lévy process; the natural class of stochastic processes which contains many examples of risk processes which have already been considered in the existing literature. Following from the important contributions of [Zhou, X., 2005. On a classical risk model with a constant dividend barrier. North Am. Act. J. 95-108] we provide an explicit characterization of a generalized version of the Gerber-Shiu function in terms of scale functions, streamlining and extending results available in the literature.

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Section: Introductionmentioning

confidence: 98%

A note on scale functions and the time value of ruin for Lévy insurance risk processes

Biffis

Kyprianou

2010

Insurance: Mathematics and Economics

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“…In [29], the author introduced a new parametric family of one-sided Lévy processes which are characterized by the following Laplace exponent, for any 1 < < 2, β ≥ 0 and γ > 1 − ,…”

Section: Lemma 34mentioning

confidence: 99%

Infinite divisibility of solutions to some self-similar integro-differential equations and exponential functionals of Lévy processes

Patie¹

2009

Ann. Inst. H. Poincaré Probab. Statist.

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We first characterize the increasing eigenfunctions associated to the following family of integro-differential operators, for any α, x > 0, γ ≥ 0 and f a smooth function on R + ,where the coefficients b ∈ R, σ ≥ 0 and the measure ν, which satisfies the integrability condition ∞ 0 (1 ∧ r 2 )ν(dr) < +∞, are uniquely determined by the distribution of a spectrally negative, infinitely divisible random variable, with characteristic exponent ψ. L (γ ) is known to be the infinitesimal generator of a positive α-self-similar Feller process, which has been introduced by Lamperti [Z. Wahrsch. Verw. Gebiete 22 (1972) 205-225]. The eigenfunctions are expressed in terms of a new family of power series which includes, for instance, the modified Bessel functions of the first kind and some generalizations of the Mittag-Leffler function. Then, we show that some specific combinations of these functions are Laplace transforms of self-decomposable or infinitely divisible distributions concentrated on the positive line with respect to the main argument, and, more surprisingly, with respect to the parameter ψ(γ ). In particular, this generalizes a result of Hartman [Ann. Sc. Norm. Super. Pisa Cl. Sci. IV-III (1976) 267-287] for the increasing solution of the Bessel differential equation. Finally, we compute, for some cases, the associated decreasing eigenfunctions and derive the Laplace transform of the exponential functionals of some spectrally negative Lévy processes with a negative first moment.Résumé. Nous commençons par caractériser les fonctions propres croissantes, au sens strict, de la famille d'opérateurs intégrodifférentiels (0.1), pour tout α > 0, γ ≥ 0, f une function définie sur R + et suffissament régulière, et où les coefficients b ∈ R, σ ≥ 0 et la mesure ν, qui satisfait la condition d'intégrabilité ∞ 0 (1 ∧ r 2 )ν(dr) < +∞, sont données, de manière unique, par la distribution d'une variable aléatoire infiniment divisible et spectralement négative dont on écrit ψ son exposant caractéristique. L (γ ) est le générateur infinitésimal d'un processus positif Fellerien α-auto-similaire, introduit par Lamperti [Z. Wahrsch. Verw. Gebiete 22 (1972) 205-225]. Les fonctions propres sont définies en terme d'une nouvelle famille de séries entières qui contient, par exemple, les fonctions de Bessel modifiées du premier ordre et des généralisations des fonctions de Mittag-Leffler. Nous continuons par montrer que des combinaisons particulières de ces séries entières correspondent à des transformées de Laplace de variables aléatoires positives auto-décomposables ou infiniment divisibles, par rapport à la valeur propre associée mais aussi par rapport au paramètre ψ(γ ), ce qui est plus surprenant. En particulier, ceci généralise un résultat de Hartman [Ann. Sc. Norm. Super. Pisa Cl. Sci. IV-III (1976) 267-287] sur les fonctions de Bessel modifiées. Finalement, nous calculons, dans certains cas, les fonctions propres décroissantes, ce qui nous permet de caractériser la loi, par le biais de sa transformée de Laplace, de la fonctionnelle...

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“…E. KYPRIANOU AND J. C. PARDO Finally, we note that ξ and ξ * are two examples of so-called Lamperti stable processes (see, for instance, [2], [4], [6], [10], and [24] for related expositions and the formal definition of a Lamperti stable process).…”

Section: E(exp{−ξ 1146mentioning

confidence: 99%

Continuous-State Branching Processes and Self-Similarity

Kyprianou

Pardo

2008

J. Appl. Probab.

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In this paper we study the α-stable continuous-state branching processes (for α ∈ (1, 2]) and the α-stable continuous-state branching processes conditioned never to become extinct in the light of positive self-similarity. Understanding the interaction of the Lamperti transformation for continuous-state branching processes and the Lamperti transformation for positive, self-similar Markov processes gives access to a number of explicit results concerning the paths of α-stable continuous-state branching processes and α-stable continuous-state branching processes conditioned never to become extinct.

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Exponential functional of a new family of Lévy processes and self-similar continuous state branching processes with immigration

Cited by 52 publications

References 31 publications

A note on scale functions and the time value of ruin for Lévy insurance risk processes

A note on scale functions and the time value of ruin for Lévy insurance risk processes

Infinite divisibility of solutions to some self-similar integro-differential equations and exponential functionals of Lévy processes

Continuous-State Branching Processes and Self-Similarity

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