Results 1  10
of
121
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 80 (6 self)
 Add to MetaCart
(Show Context)
Abstract: This text surveys properties and applications of the exponential functional ∫ t exp(−ξs)ds of realvalued Lévy processes ξ = (ξt, t ≥ 0). 0
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 66 (2 self)
 Add to MetaCart
(Show Context)
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.
Tail asymptotics for exponential functionals of Lévy processes
 Stochastic Processes and their Applications 116 (2), 156–177.s and Economics 46 (2010) 362–370
, 2006
"... Motivated by recent studies in financial mathematics and other areas, we investigate the exponential functional Z = 0 e−X(t)dt of a Lévy process X(t), t ≥ 0. In particular, we investigate its tail asymptotics. We show that, depending on the right tail of X(1), the tail behavior of Z is exponential ..."
Abstract

Cited by 62 (5 self)
 Add to MetaCart
(Show Context)
Motivated by recent studies in financial mathematics and other areas, we investigate the exponential functional Z = 0 e−X(t)dt of a Lévy process X(t), t ≥ 0. In particular, we investigate its tail asymptotics. We show that, depending on the right tail of X(1), the tail behavior of Z is exponential, Pareto, or extremely heavytailed.
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 55 (6 self)
 Add to MetaCart
(Show Context)
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 ..."
(Show Context)
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 40 (21 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].
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 38 (9 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].
Spectral Expansions for Asian (Average Price) Options
, 2004
"... Arithmetic Asian or average price options deliver payoffs based on the average underlying price over a prespecified time period. Asian options are an important family of derivative contracts with a wide variety of applications in currency, equity, interest rate, commodity, energy, and insurance mark ..."
Abstract

Cited by 32 (4 self)
 Add to MetaCart
Arithmetic Asian or average price options deliver payoffs based on the average underlying price over a prespecified time period. Asian options are an important family of derivative contracts with a wide variety of applications in currency, equity, interest rate, commodity, energy, and insurance markets. We derive two analytical formulas for the value of the continuously sampled arithmetic Asian option when the underlying asset price follows geometric Brownian motion. We use an identity in law between the integral of geometric Brownian motion over a finite time interval 0 t and the state at time t of a onedimensional diffusion process with affine drift and linear diffusion and express Asian option values in terms of spectral expansions associated with the diffusion infinitesimal generator. The first formula is an infinite series of terms involving Whittaker functions M and W. The second formula is a single real integral of an expression involving Whittaker function W plus (for some parameter values) a finite number of additional terms involving incomplete gamma functions and Laguerre polynomials. The two formulas allow accurate computation of continuously sampled arithmetic Asian option prices.
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 31 (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.