Results 1  10
of
22
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 ..."
Abstract

Cited by 98 (9 self)
 Add to MetaCart
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.
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
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 ..."
Continuous time volatility modelling: COGARCH versus OrnsteinUhlenbeck models
 FROM STOCHASTIC CALCULUS TO MATHEMATICAL FINANCE. THE SHIRYAEV FESTSCHRIFT
, 2006
"... We compare the probabilistic properties of the nonGaussian OrnsteinUhlenbeck based stochastic volatility model of BarndorffNielsen and Shephard (2001) with those of the COGARCH process. The latter is a continuous time GARCH process introduced by the authors (2004). Many features are shown to be ..."
Abstract

Cited by 26 (12 self)
 Add to MetaCart
We compare the probabilistic properties of the nonGaussian OrnsteinUhlenbeck based stochastic volatility model of BarndorffNielsen and Shephard (2001) with those of the COGARCH process. The latter is a continuous time GARCH process introduced by the authors (2004). Many features are shown to be shared by both processes, but differences are pointed out as well. Furthermore, it is shown that the COGARCH process has Pareto like tails under weak regularity conditions.
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.
A law of iterated logarithm for increasing selfsimilar Markov processes
, 2002
"... We consider increasing selfsimilar Markov processes (X t , t 0) on ]0, #[. ..."
Abstract

Cited by 11 (4 self)
 Add to MetaCart
We consider increasing selfsimilar Markov processes (X t , t 0) on ]0, #[.
A transformation from Hausdorff to Stieltjes moment sequences
 Ark. Mat
, 2004
"... Abstract We introduce a nonlinear injective transformation T from the set of nonvanishing normalized Hausdorff moment sequences to the set of normalized Stieltjes moment sequences by the formula T [(an)]n = 1/(a1 *... * an). Special cases of this transformation have appeared in various papers on e ..."
Abstract

Cited by 10 (6 self)
 Add to MetaCart
Abstract We introduce a nonlinear injective transformation T from the set of nonvanishing normalized Hausdorff moment sequences to the set of normalized Stieltjes moment sequences by the formula T [(an)]n = 1/(a1 *... * an). Special cases of this transformation have appeared in various papers on exponential functionals of L'evy processes, partly motivated by mathematical finance. We give several examples of moment sequences arising from the transformation and provide the corresponding measures, some of which are related to qseries. 2000 Mathematics Subject Classification: primary 44A60; secondary 33D65. Keywords: moment sequence, qseries. 1 Introduction and main results In his fundamental memoir [23] Stieltjes characterized sequences of the form sn = Z 1
A TRANSFORMATION FOR LÉVY PROCESSES WITH ONESIDED JUMPS AND APPLICATIONS
"... Abstract. The aim of this work is to extend and study a family of transformations between Laplace exponents of Lévy processes which have been introduced recently in a variety of different contexts, [27, 29, 21, 17], as well as in older work of Urbanik [35]. We show how some specific instances of thi ..."
Abstract

Cited by 5 (3 self)
 Add to MetaCart
Abstract. The aim of this work is to extend and study a family of transformations between Laplace exponents of Lévy processes which have been introduced recently in a variety of different contexts, [27, 29, 21, 17], as well as in older work of Urbanik [35]. We show how some specific instances of this mapping prove to be useful for a variety of applications. 1.