Results 1  10
of
58
Exponential functionals of Lévy processes
 Probabilty Surveys
, 2005
"... Abstract: This text surveys properties and applications of the exponential functional ∫ t exp(−ξs)ds of realvalued Lévy processes ξ = (ξt, t ≥ 0). 0 ..."
Abstract

Cited by 34 (4 self)
 Add to MetaCart
Abstract: This text surveys properties and applications of the exponential functional ∫ t exp(−ξs)ds of realvalued Lévy processes ξ = (ξt, t ≥ 0). 0
Regenerative composition structures
 ANN. PROBAB
, 2005
"... A new class of random composition structures (the ordered analog of Kingman’s partition structures) is defined by a regenerative description of component sizes. Each regenerative composition structure is represented by a process of random sampling of points from an exponential distribution on the po ..."
Abstract

Cited by 32 (18 self)
 Add to MetaCart
A new class of random composition structures (the ordered analog of Kingman’s partition structures) is defined by a regenerative description of component sizes. Each regenerative composition structure is represented by a process of random sampling of points from an exponential distribution on the positive halfline, and separating the points into clusters by an independent regenerative random set. Examples are composition structures derived from residual allocation models, including one associated with the Ewens sampling formula, and composition structures derived from the zero set of a Brownian motion or Bessel process. We provide characterisation results and formulas relating the distribution of the regenerative composition to the Lévy parameters of a subordinator whose range is the corresponding regenerative set. In particular, the only reversible regenerative composition structures are those associated with the interval partition of [0, 1] generated by excursions of a standard Bessel bridge of dimension 2 − 2α for some α ∈ [0, 1].
AIMD algorithms and exponential functionals
 Ann. Appl. Probab
, 2002
"... ABSTRACT. The behavior of connection transmitting packets into a network according to a general additiveincrease multiplicativedecrease (AIMD) algorithm is investigated. It is assumed that loss of packets occurs in clumps. When a packet is lost, a certain number of subsequent packets are also lost ..."
Abstract

Cited by 31 (4 self)
 Add to MetaCart
ABSTRACT. The behavior of connection transmitting packets into a network according to a general additiveincrease multiplicativedecrease (AIMD) algorithm is investigated. It is assumed that loss of packets occurs in clumps. When a packet is lost, a certain number of subsequent packets are also lost (correlated losses). The stationary behavior of this algorithm is analyzed when the rate of occurrence of clumps becomes arbitrarily small. From a probabilistic point of view, it is shown that exponential functionals associated to compound Poisson processes play a key role. A formula for the fractional moments and some density functions are derived. Analytically, to get the explicit expression of the distributions involved, the natural framework of this study turns out to be the qcalculus. Different loss models are then compared using concave ordering. Quite surprisingly, it is shown that, for a fixed loss rate, the correlated loss model has a higher throughput than an uncorrelated loss model. CONTENTS
The genealogy of selfsimilar fragmentations with negative index as a continuum random tree
 Electr. J. Prob
, 2004
"... continuum random tree ..."
Homogeneous fragmentation processes
, 2000
"... The purpose of this work is to define and study homogeneous fragmentation processes in continuous time, which are meant to describe the evolution of an object that breaks down randomly into pieces as time passes. Roughly, we show that the dynamic of such a fragmentation process is determined by som ..."
Abstract

Cited by 26 (4 self)
 Add to MetaCart
The purpose of this work is to define and study homogeneous fragmentation processes in continuous time, which are meant to describe the evolution of an object that breaks down randomly into pieces as time passes. Roughly, we show that the dynamic of such a fragmentation process is determined by some exchangeable measure on the set of partitions of N, and results from the combination of two different phenomena: a continuous erosion and sudden dislocations. In particular, we determine the class of fragmentation measures which can arise in this setting, and investigate the evolution of the size of the fragment that contains a point pick at random at the initial time.
Asymptotic laws for compositions derived from transformed subordinators
 ANN. PROBAB
, 2006
"... A random composition of n appears when the points of a random closed set ˜ R ⊂ [0, 1] are used to separate into blocks n points sampled from the uniform distribution. We study the number of parts Kn of this composition and other related functionals under the assumption that ˜ R = φ(S•) where (St, t ..."
Abstract

Cited by 25 (10 self)
 Add to MetaCart
A random composition of n appears when the points of a random closed set ˜ R ⊂ [0, 1] are used to separate into blocks n points sampled from the uniform distribution. We study the number of parts Kn of this composition and other related functionals under the assumption that ˜ R = φ(S•) where (St, t ≥ 0) is a subordinator and φ: [0, ∞] → [0, 1] is a diffeomorphism. We derive the asymptotics of Kn when the Lévy measure of the subordinator is regularly varying at 0 with positive index. Specialising to the case of exponential function φ(x) = 1 −e −x we establish a connection between the asymptotics of Kn and the exponential functional of the subordinator.
Selfsimilar fragmentations
, 2000
"... We introduce a probabilistic model that is meant to describe an object that falls apart randomly as time passes and fulfills a certain scaling property. We show that the distribution of such a process is determined by its index of selfsimilarity α ∈ R, a rate of erosion c ≥ 0, and a socalled Lév ..."
Abstract

Cited by 24 (8 self)
 Add to MetaCart
We introduce a probabilistic model that is meant to describe an object that falls apart randomly as time passes and fulfills a certain scaling property. We show that the distribution of such a process is determined by its index of selfsimilarity α ∈ R, a rate of erosion c ≥ 0, and a socalled Lévy measure that accounts for sudden dislocations. The key of the analysis is provided by a transformation of selfsimilar fragmentations which enables us to reduce the study to the homogeneous case α = 0 which is treated in [6].
Recurrent extensions of selfsimilar Markov processes and Cramér’s condition
 Bernoulli
, 2005
"... We prove that a positive selfsimilar Markov process (X,P) that hits 0 in a finite time admits a selfsimilar recurrent extension that leaves 0 continuously if and only if the underlying Lévy process satisfies Cramér’s condition. ..."
Abstract

Cited by 23 (2 self)
 Add to MetaCart
We prove that a positive selfsimilar Markov process (X,P) that hits 0 in a finite time admits a selfsimilar recurrent extension that leaves 0 continuously if and only if the underlying Lévy process satisfies Cramér’s condition.
Exponential functionals of Lévy processes
"... 23> ; with E \Theta e t = e \GammatOE() 3. Martingale methods. The perpetuity equation Let T be a finite stopping time. Then t = t+T \Gamma t , j t = j t+T \Gamma j t are L'evy processes independent of F T . Since A1 (; j) = A T (; j) + e T A1 ( ; j); 1 therefore A1 is a solut ..."
Abstract

Cited by 21 (1 self)
 Add to MetaCart
23> ; with E \Theta e t = e \GammatOE() 3. Martingale methods. The perpetuity equation Let T be a finite stopping time. Then t = t+T \Gamma t , j t = j t+T \Gamma j t are L'evy processes independent of F T . Since A1 (; j) = A T (; j) + e T A1 ( ; j); 1 therefore A1 is a solution of the perpetuity equation X d =UX + V X?(U;V ):
On Continuity Properties of the Law of Integrals of Lévy Processes
, 2008
"... Let (ξ,η) be a bivariate Lévy process such that the integral ∫ ∞ 0 e−ξt − dηt converges almost surely. We characterise, in terms of their Lévy measures, those Lévy processes for which (the distribution of) this integral has atoms. We then turn attention to almost surely convergent integrals of the f ..."
Abstract

Cited by 14 (4 self)
 Add to MetaCart
Let (ξ,η) be a bivariate Lévy process such that the integral ∫ ∞ 0 e−ξt − dηt converges almost surely. We characterise, in terms of their Lévy measures, those Lévy processes for which (the distribution of) this integral has atoms. We then turn attention to almost surely convergent integrals of the form I: = ∫ ∞ 0 g(ξt)dt, where g is a deterministic function. We give sufficient conditions ensuring that I has no atoms, and under further conditions derive that I has a Lebesgue density. The results are also extended to certain integrals of the form ∫ ∞ 0 g(ξt)dYt, where Y is an almost surely strictly increasing stochastic process, independent of ξ.