Displaying similar documents to “On suprema of Lévy processes and application in risk theory”

Refracted Lévy processes

A. E. Kyprianou, R. L. Loeffen (2010)

Annales de l'I.H.P. Probabilités et statistiques

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Motivated by classical considerations from risk theory, we investigate boundary crossing problems for refracted Lévy processes. The latter is a Lévy process whose dynamics change by subtracting off a fixed linear drift (of suitable size) whenever the aggregate process is above a pre-specified level. More formally, whenever it exists, a refracted Lévy process is described by the unique strong solution to the stochastic differential equation d =− { ...

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

Patie Pierre (2009)

Annales de l'I.H.P. Probabilités et statistiques

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We first characterize the increasing eigenfunctions associated to the following family of integro-differential operators, for any , >0, ≥0 and a smooth function on ℜ + , 𝐋 ( γ ) f ( x ) = x - α ( σ 2 x 2 f ' ' ( x ) + ( σ γ + b ) x f ' ( x ) + ∫ 0 ∞ f e - r x - f ( x ) e - r γ + x f ' ( x ) r 𝕀 { r ≤ 1 } ν ( d r ) ) , ( 0 . 1 ) where the coefficients b ∈ ℜ ,≥0 and the measure , which satisfies the integrability condition (1∧ )(d)<+∞, are uniquely determined by the distribution of a spectrally negative, infinitely divisible random variable, with characteristic exponent . ...

Small-time behavior of beta coalescents

Julien Berestycki, Nathanaël Berestycki, Jason Schweinsberg (2008)

Annales de l'I.H.P. Probabilités et statistiques

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For a finite measure on [0, 1], the -coalescent is a coalescent process such that, whenever there are clusters, each -tuple of clusters merges into one at rate (1−) (d). It has recently been shown that if 1<<2, the -coalescent in which is the Beta (2−, ) distribution can be used to describe the genealogy of a continuous-state branching process (CSBP) with an -stable branching mechanism. Here...

A Ciesielski–Taylor type identity for positive self-similar Markov processes

A. E. Kyprianou, P. Patie (2011)

Annales de l'I.H.P. Probabilités et statistiques

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The aim of this note is to give a straightforward proof of a general version of the Ciesielski–Taylor identity for positive self-similar Markov processes of the spectrally negative type which umbrellas all previously known Ciesielski–Taylor identities within the latter class. The approach makes use of three fundamental features. Firstly, a new transformation which maps a subset of the family of Laplace exponents of spectrally negative Lévy processes into itself. Secondly, some classical...