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105
Exponential functionals of Lévy processes
- Probabilty Surveys
, 2005
"... Abstract: This text surveys properties and applications of the exponential functional ∫ t exp(−ξs)ds of real-valued Lévy processes ξ = (ξt, t ≥ 0). 0 ..."
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Cited by 76 (6 self)
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Abstract: This text surveys properties and applications of the exponential functional ∫ t exp(−ξs)ds of real-valued Lévy processes ξ = (ξt, t ≥ 0). 0
Recurrent extensions of self-similar Markov processes and Cramér’s condition
- Bernoulli
, 2005
"... We prove that a positive self-similar Markov process (X,P) that hits 0 in a finite time admits a self-similar recurrent extension that leaves 0 continuously if and only if the underlying Lévy process satisfies Cramér’s condition. ..."
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Cited by 56 (2 self)
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We prove that a positive self-similar Markov process (X,P) that hits 0 in a finite time admits a self-similar recurrent extension that leaves 0 continuously if and only if the underlying Lévy process satisfies Cramér’s condition.
AIMD algorithms and exponential functionals
- Ann. Appl. Probab
, 2002
"... ABSTRACT. The behavior of connection transmitting packets into a network according to a general additive-increase multiplicative-decrease (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 ..."
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Cited by 56 (6 self)
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ABSTRACT. The behavior of connection transmitting packets into a network according to a general additive-increase multiplicative-decrease (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 q-calculus. 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
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 investi-gate 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 ..."
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Cited by 53 (5 self)
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Motivated by recent studies in financial mathematics and other areas, we investi-gate 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 heavy-tailed.
The genealogy of self-similar fragmentations with negative index as a continuum random tree
- Electr. J. Prob
, 2004
"... continuum random tree ..."
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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 ..."
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Cited by 39 (22 self)
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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].
Exponential functionals of Lévy processes. In:
- Lévy Processes, Theory and Applications,
, 2001
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Self-similar 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 self-similarity α ∈ R, a rate of erosion c ≥ 0, and a so-called Lév ..."
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Cited by 37 (9 self)
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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 self-similarity α ∈ R, a rate of erosion c ≥ 0, and a so-called Lévy measure that accounts for sudden dislocations. The key of the analysis is provided by a transformation of self-similar 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 ..."
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Cited by 32 (4 self)
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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 one-dimensional 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.
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 ..."
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Cited by 29 (12 self)
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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.
