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55
The twoparameter PoissonDirichlet distribution derived from a stable subordinator.
, 1995
"... The twoparameter PoissonDirichlet distribution, denoted pd(ff; `), is a distribution on the set of decreasing positive sequences with sum 1. The usual PoissonDirichlet distribution with a single parameter `, introduced by Kingman, is pd(0; `). Known properties of pd(0; `), including the Markov ..."
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Cited by 221 (37 self)
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The twoparameter PoissonDirichlet distribution, denoted pd(ff; `), is a distribution on the set of decreasing positive sequences with sum 1. The usual PoissonDirichlet distribution with a single parameter `, introduced by Kingman, is pd(0; `). Known properties of pd(0; `), including the Markov chain description due to VershikShmidtIgnatov, are generalized to the twoparameter case. The sizebiased random permutation of pd(ff; `) is a simple residual allocation model proposed by Engen in the context of species diversity, and rediscovered by Perman and the authors in the study of excursions of Brownian motion and Bessel processes. For 0 ! ff ! 1, pd(ff; 0) is the asymptotic distribution of ranked lengths of excursions of a Markov chain away from a state whose recurrence time distribution is in the domain of attraction of a stable law of index ff. Formulae in this case trace back to work of Darling, Lamperti and Wendel in the 1950's and 60's. The distribution of ranked lengths of e...
On some exponential functionals of Brownian motion
 Adv. Appl. Prob
, 1992
"... Abstract: This is the second part of our survey on exponential functionals of Brownian motion. We focus on the applications of the results about the distributions of the exponential functionals, which have been discussed in the first part. Pricing formula for call options for the Asian options, expl ..."
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Cited by 98 (9 self)
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Abstract: This is the second part of our survey on exponential functionals of Brownian motion. We focus on the applications of the results about the distributions of the exponential functionals, which have been discussed in the first part. Pricing formula for call options for the Asian options, explicit expressions for the heat kernels on hyperbolic spaces, diffusion processes in random environments and extensions of Lévy’s and Pitman’s theorems are discussed.
On The Distribution And Asymptotic Results For Exponential Functionals Of Lévy Processes
, 1997
"... . The aim of this note is to study the distribution and the asymptotic behavior of the exponential functional A t := R t 0 e s ds, where ( s ; s 0) denotes a L'evy process. When A1 ! 1, we show that in most cases, the law of A1 is a solution of an integrodifferential equation ; moreover, this ..."
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Cited by 65 (8 self)
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. The aim of this note is to study the distribution and the asymptotic behavior of the exponential functional A t := R t 0 e s ds, where ( s ; s 0) denotes a L'evy process. When A1 ! 1, we show that in most cases, the law of A1 is a solution of an integrodifferential equation ; moreover, this law is characterized by its integral moments. When the process is asymptotically ffstable, we prove that t \Gamma1=ff log A t converges in law, as t !1, to the supremum of an ffstable L'evy process ; in particular, if E [ 1 ] ? 0, then ff = 1 and (1=t) log A t converges almost surely to E [ 1 ]. Eventually, we use Girsanov's transform to give the explicit behavior of E \Theta (a +A t ()) \Gamma1 as t ! 1, where a is a constant, and deduce from this the rate of decay of the tail of the distribution of the maximum of a diffusion process in a random L'evy environment. 1. Introduction We first describe three different sources of interest for exponential functionals of Brownian mot...
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 ..."
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Cited by 34 (4 self)
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Abstract: This text surveys properties and applications of the exponential functional ∫ t exp(−ξs)ds of realvalued Lévy processes ξ = (ξt, t ≥ 0). 0
A Survey and Some Generalizations of Bessel Processes
 Bernoulli
, 1999
"... Bessel processes play an important role in financial mathematics because of their strong relation to financial processes like geometric Brownian motion or CIR processes. We are interested in the first time Bessel processes and more generally, radial OrnsteinUhlenbeck processes hit a given barrier. ..."
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Cited by 27 (1 self)
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Bessel processes play an important role in financial mathematics because of their strong relation to financial processes like geometric Brownian motion or CIR processes. We are interested in the first time Bessel processes and more generally, radial OrnsteinUhlenbeck processes hit a given barrier. We give explicit expressions of the Laplace transforms of first hitting times by (squared) radial OrnsteinUhlenbeck processes, i. e., CIR processes. As a natural extension we study squared Bessel processes and squared OrnsteinUhlenbeck processes with negative dimensions or negative starting points and derive their properties. Keywords: First hitting times; CIR processes; Bessel processes; radial Ornstein Uhlenbeck processes; Bessel processes with negative dimensions 1 Introduction Bessel processes have come to play a distinguished role in financial mathematics for at least two reasons, which have a lot to do with the models being usually considered. One of these models is the CoxI...
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. ..."
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Cited by 23 (2 self)
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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.
Weak convergence of positive self similar Markov processes and overshoots of Lévy processes
, 2004
"... We give necessary and sufficient conditions for the law of a positive selfsimilar Markov process to converge weakly as its initial state tends to 0 and we describe the limit law. Our proof is based on Lamperti's representation which relates any positive selfsimilar process to a unique Levy process ..."
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Cited by 14 (2 self)
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We give necessary and sufficient conditions for the law of a positive selfsimilar Markov process to converge weakly as its initial state tends to 0 and we describe the limit law. Our proof is based on Lamperti's representation which relates any positive selfsimilar process to a unique Levy process. Then we show that the convergence mentioned above holds if and only if the process of the overshoots of the underlying Levy process # in the Lamperti's representation converges weakly at infinity and E T1 where T 1 = inf{t : # t 1}. Under these conditions, we give a pathwise construction of the limit law.
Regenerative tree growth: binary selfsimilar continuum random trees and PoissonDirichlet compositions
, 2008
"... We use a natural ordered extension of the Chinese Restaurant Process to grow a twoparameter family of binary selfsimilar continuum fragmentation trees. We provide an explicit embedding of Ford’s sequence of alpha model trees in the continuum tree which we identified in a previous article as a dist ..."
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Cited by 14 (7 self)
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We use a natural ordered extension of the Chinese Restaurant Process to grow a twoparameter family of binary selfsimilar continuum fragmentation trees. We provide an explicit embedding of Ford’s sequence of alpha model trees in the continuum tree which we identified in a previous article as a distributional scaling limit of Ford’s trees. In general, the Markov branching trees induced by the twoparameter growth rule are not sampling consistent, so the existence of compact limiting trees cannot be deduced from previous work on the sampling consistent case. We develop here a new approach to establish such limits, based on regenerative interval partitions and the urnmodel description of sampling from Dirichlet random distributions. 1. Introduction. We
Asymptotic laws for nonconservative selfsimilar fragmentations
 Electronic J. Probab
, 2004
"... Abstract We consider a selfsimilar fragmentation process in which the generic particle of size x is replaced at probability rate x α by its offspring made of smaller particles, where α is some positive parameter. The total of offspring sizes may be both larger or smaller than x with positive probab ..."
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Cited by 14 (2 self)
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Abstract We consider a selfsimilar fragmentation process in which the generic particle of size x is replaced at probability rate x α by its offspring made of smaller particles, where α is some positive parameter. The total of offspring sizes may be both larger or smaller than x with positive probability. We show that under certain conditions the typical size in the ensemble is of the order t −1/α and that the empirical distribution of sizes converges to a random limit which we characterise in terms of the reproduction law. 1