## Exponential functionals of Lévy processes (2005)

Venue: | Probabilty Surveys |

Citations: | 34 - 4 self |

### BibTeX

@INPROCEEDINGS{Bertoin05exponentialfunctionals,

author = {Jean Bertoin and Marc Yor},

title = {Exponential functionals of Lévy processes},

booktitle = {Probabilty Surveys},

year = {2005},

pages = {191--212}

}

### OpenURL

### Abstract

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

### Citations

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Lévy processes and infinitely divisible distributions (english ed.)., volume 68 of Cambridge studies in advanced mathematics
- Sato
- 2005
(Show Context)
Citation Context ...ications: Primary 60 G 51, 60 J 55; secondary 60 G 18, 44 A 60. Received December 2004. 1. Introduction Throughout this work, we shall consider a real-valued Lévy process ξ = (ξt, t ≥ 0), refering to =-=[5, 60]-=- for background. This means that the process ξ starts from ξ0 = 0, has right-continuous paths with left-limits, and if (Ft)t≥0 denotes the natural filtration generated by ξ, then the increment ξt+s −ξ... |

647 |
Basic Hypergeometric Series
- Gasper, Rahman
- 1990
(Show Context)
Citation Context ...) is a sequence of i.i.d. exponential variables with parameter 1. In order to specify the distribution of the exponential functional, it is quite convenient to use the so-called q-calculus (see, e.g. =-=[31]-=-, [44], ...) which is associated with the basic hypergeometric series of Euler, Gauss, ... More precisely, let us introduce some standard notation in this setting: and n−1 ∏ (a; q)n = (1 − aq j ) , (a... |

413 |
Lévy Processes
- Bertoin
- 1996
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Citation Context ...ications: Primary 60 G 51, 60 J 55; secondary 60 G 18, 44 A 60. Received December 2004. 1. Introduction Throughout this work, we shall consider a real-valued Lévy process ξ = (ξt, t ≥ 0), refering to =-=[5, 60]-=- for background. This means that the process ξ starts from ξ0 = 0, has right-continuous paths with left-limits, and if (Ft)t≥0 denotes the natural filtration generated by ξ, then the increment ξt+s −ξ... |

161 |
Random difference equations and renewal theory for products of random matrices
- Kesten
(Show Context)
Citation Context ...les I∞ and (A, B) are independent. Specifically, the law of the latter pair is given by : (A, B) L ( ∫ ) T = exp(−ξT), exp(−ξs)ds . The random affine equation has been investigated in depth by Kesten =-=[45]-=-; see also Vervaat [66] and Goldie [35]. An interesting application of the results of Kesten and Goldie in the special case when we take T ≡ 1 is the following estimation of the tail distribution of t... |

127 | An Introduction to Probability Theory and Its Applications, 2nd ed - Feller - 1957 |

126 |
Implicit renewal theory and tails of solutions of random equations
- Goldie
- 1991
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Citation Context ...ifically, the law of the latter pair is given by : (A, B) L ( ∫ ) T = exp(−ξT), exp(−ξs)ds . The random affine equation has been investigated in depth by Kesten [45]; see also Vervaat [66] and Goldie =-=[35]-=-. An interesting application of the results of Kesten and Goldie in the special case when we take T ≡ 1 is the following estimation of the tail distribution of the exponential functional under a condi... |

97 | Exponential Functionals of Brownian Motion
- Yor
- 2001
(Show Context)
Citation Context ... adopted between 1973 and 1990, say), and later generalized to become the exponential of Lévy processes (ξt, t ≥ 0). In particular, the computation of the price of Asian options (see, e.g., papers in =-=[70]-=-) is equivalent to the knowledge of the law of ∫ t 0 exp(−ξs)ds, for fixed t; an important reduction of the problem consists in replacing t by an exponential time θ, which is independent of ξ (in othe... |

88 | The classical moment problem as a self-adjoint finite difference operator
- Simon
- 1998
(Show Context)
Citation Context ...is a Poisson process, and more precisely, that the negative entire moments of the exponential functional of a Poisson process are closely related to those of the log-normal distribution). We refer to =-=[62, 64]-=- and the references therein for more about indeterminacy for the moment problem. 4. On the distribution of the exponential functional 4.1. Some important special cases In this section, we briefly pres... |

83 |
On a stochastic difference equation and a representation of non-negative infinitely divisible random variables
- Vervaat
- 1979
(Show Context)
Citation Context ...ndependent. Specifically, the law of the latter pair is given by : (A, B) L ( ∫ ) T = exp(−ξT), exp(−ξs)ds . The random affine equation has been investigated in depth by Kesten [45]; see also Vervaat =-=[66]-=- and Goldie [35]. An interesting application of the results of Kesten and Goldie in the special case when we take T ≡ 1 is the following estimation of the tail distribution of the exponential function... |

77 |
Quantum Calculus
- Kac, Cheung
- 2002
(Show Context)
Citation Context ... sequence of i.i.d. exponential variables with parameter 1. In order to specify the distribution of the exponential functional, it is quite convenient to use the so-called q-calculus (see, e.g. [31], =-=[44]-=-, ...) which is associated with the basic hypergeometric series of Euler, Gauss, ... More precisely, let us introduce some standard notation in this setting: and n−1 ∏ (a; q)n = (1 − aq j ) , (a; q)∞ ... |

73 |
The distribution of a perpetuity, with applications to risk theory and pension funding
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- 1990
(Show Context)
Citation Context ...f 2γb, where γb is a gamma variable with index b. Hence Theorem 3 enables us to recover the identity in distribution ∫ ∞ exp {−2(Bs + bs)} ds L = 1 , (16) 2γb 0 which has been established by Dufresne =-=[26]-=- (see also Proposition 3 in PollakSiegmund [57], Example 3.3 on page 309 in Urbanik [65], Yor [68], ...). A further discussion of (16) in relation with DNA is made in [46] and [48]. • Exponential Lévy... |

65 | On the distribution and asymptotic results for exponential functionals of Lévy processes. In Exponential functionals and principal values related to Brownian motion
- Carmona, Petit, et al.
- 1997
(Show Context)
Citation Context ...rtant special cases In this section, we briefly present some explicit results on the distribution of the exponential functional in three important special cases; further examples can be found e.g. in =-=[20, 21]-=-. • Standard Poisson process: Here, we consider a standard Poisson process (Nt, t ≥ 0) and define for q ∈]0, 1[ its exponential functional by I (q) ∞ = ∫ ∞ q Nt dt . 0In other words, I (q) ∞ I (q) ∞ ... |

57 |
Semi-stable Markov processes
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Citation Context ...situations where Cramer’s condition is not fulfilled. 0 0J. Bertoin and M. Yor/Exponential functionals 202 5. Self-similar Markov processes Motivated by limit theorems for Markov processes, Lamperti =-=[47]-=- considered families of probability measures (Px, x > 0) on Skorohod’s space D of càdlàg paths ω : R+ → R+, under which the coordinate process X·(ω) = ω(·) becomes Markovian and fulfills the scaling p... |

52 | Diffusions, Markov processes, and martingales, Volume 1 of Wiley series in probability and mathematical statistics - Rogers, Williams - 1994 |

31 | AIMD algorithms and exponential functionals - Guillemin, Robert, et al. |

30 |
On subordinators, self-similar Markov processes and some factorizations of the exponential variable. Electron
- Bertoin, Yor
- 2001
(Show Context)
Citation Context ...eorem 2(ii) also invites to look for factorizations of an exponential variable in two independent factors, one of which may be obtained (in distribution) as the exponential functional I∞. We refer to =-=[13]-=- for several classical such factorizations which fit in this setting. We conclude this section by mentioning that moments calculations as in Theorem 2(i) can be extended to the situation when ξ is an ... |

30 | Regenerative composition structures - Gnedin, Pitman - 2005 |

27 |
The asymptotic behavior of fragmentation processes
- BERTOIN
- 2003
(Show Context)
Citation Context ...tion process. In particular, this approach yields important limit theorems for the empirical distribution of the fragments of self-similar fragmentations with a positive index of self-similarity; see =-=[9, 12, 17, 30]-=-. On the other hand, self-similar fragmentations with a negative index of self-similarity are dissipative, in the sense that the total mass of the fragments decreases as time passes. In this direction... |

24 | Random Fragmentation and - Bertoin - 2006 |

24 | Recurrent extensions of self-similar Markov processes and Cramér’s condition
- Rivero
- 2005
(Show Context)
Citation Context ... number r > 0 such that P(ξt ∈ rZ) = 1 for all t ≥ 0). Then there exists some constant c > 0 such that P(I∞ > t) ∼ ct −θ , t → ∞ . This interesting observation has been made by Méjane [53] and Rivero =-=[58]-=-. It constrasts with the situation when ξ is a subordinator (then, obviously, the condition of Corollary 5 never holds), as we know that then I∞ possesses some finite exponential moments. Finally, we ... |

23 |
On the entire moments of self-similar Markov processes and exponential functionals of Lévy processes. Annales de la Faculté des Sciences de Toulouse Mathématiques (Série 6
- J, Yor
- 2002
(Show Context)
Citation Context ... −qx )Π(dx), q ≥ 0. ]0,∞[ We refer to [7] for background on subordinators. The first part of the following theorem has been established by Carmona et al. [20], the second is an immediate extension of =-=[15]-=-. Theorem 2 Suppose that ξ is a subordinator with Laplace exponent Φ. Let T denote a random time which has an exponential distribution with parameter q ≥ 0 (for q = 0, we have T ≡ ∞) and is independen... |

23 | Asymptotic laws for compositions derived from transformed subordinators - Gnedin, Pitman, et al. |

23 | Some aspects of Brownian motion, Part II: Some recent martingale problems - Yor - 1997 |

21 | Exponential functionals of Levy processes - Carmona, Petit, et al. - 2001 |

21 |
2006. Tail asymptotics of exponential functionals of Lévy processes. Stochastic Processes and their Applications
- Maulik, Zwart
(Show Context)
Citation Context ...situation when ξ is a subordinator (then, obviously, the condition of Corollary 5 never holds), as we know that then I∞ possesses some finite exponential moments. Finally, we refer to the recent work =-=[52]-=- for extensions of Corollary 5 to situations where Cramer’s condition is not fulfilled. 0 0J. Bertoin and M. Yor/Exponential functionals 202 5. Self-similar Markov processes Motivated by limit theore... |

20 |
The entrance laws of self-similar Markov processes and exponential functionals of Lévy processes
- Bertoin, Yor
(Show Context)
Citation Context ...lf-similar Markov process tends to ∞ without reaching ∞, and it is therefore interesting to obtain sharper information about the asymptotic behavior of Xt as t → ∞. The following result was proved in =-=[14]-=- (see also Bertoin and Caballero [11] in the special case when the Lévy process ξ is a subordinator). Theorem 6 Let X denote a self-similar Markov process with index α > 0 associated by Lamperti’s tra... |

20 |
Loss of mass in deterministic and random fragmentations, Stochastic Process
- Haas
(Show Context)
Citation Context ...process, i.e. the time at which the mass of the tagged fragment vanishes. Distributional properties of the exponential functional provide a crucial tool for the analysis of the loss of mass; see Haas =-=[37, 38]-=-. In a related area, we mention that the terminal value of the exponential functional of subordinators also arises in the study of regenerative compositions, see the recent work of Gnedin, Pitman and ... |

19 |
Entrance from 0+ for increasing semi-stable Markov processes
- Bertoin, Caballero
(Show Context)
Citation Context ...without reaching ∞, and it is therefore interesting to obtain sharper information about the asymptotic behavior of Xt as t → ∞. The following result was proved in [14] (see also Bertoin and Caballero =-=[11]-=- in the special case when the Lévy process ξ is a subordinator). Theorem 6 Let X denote a self-similar Markov process with index α > 0 associated by Lamperti’s transformation to the Lévy process ξ. Su... |

19 |
A diffusion process and its applications to detecting a change in the drift of Brownian motion”. Biometrika 72:267–280
- Pollak, Siegmund
- 1985
(Show Context)
Citation Context ...b. Hence Theorem 3 enables us to recover the identity in distribution ∫ ∞ exp {−2(Bs + bs)} ds L = 1 , (16) 2γb 0 which has been established by Dufresne [26] (see also Proposition 3 in PollakSiegmund =-=[57]-=-, Example 3.3 on page 309 in Urbanik [65], Yor [68], ...). A further discussion of (16) in relation with DNA is made in [46] and [48]. • Exponential Lévy measure: Suppose ξ has bounded variation with ... |

18 |
Self-similar processes with independent increments
- Sato
- 1991
(Show Context)
Citation Context ...ng additive process, i.e. its increments are independent but not necessarily homogeneous. This situation arises naturally in particular in the representation of self-decomposable laws on R+ (see Sato =-=[59]-=- and Jeanblanc, Pitman and Yor [41]). It also occurs in in non-parametric Bayesian statistics, where the mean of a random distribution chosen from a neutral-to-the-right prior can be represented as th... |

17 |
certaines fonctionnelles exponentielles du mouvement brownien réel
- Yor
- 1992
(Show Context)
Citation Context ...ty in distribution ∫ ∞ exp {−2(Bs + bs)} ds L = 1 , (16) 2γb 0 which has been established by Dufresne [26] (see also Proposition 3 in PollakSiegmund [57], Example 3.3 on page 309 in Urbanik [65], Yor =-=[68]-=-, ...). A further discussion of (16) in relation with DNA is made in [46] and [48]. • Exponential Lévy measure: Suppose ξ has bounded variation with drift coefficient +1 and Lévy measure Π(dx) = (a + ... |

16 |
Risk theory in a stochastic economic environment
- Paulsen
- 1993
(Show Context)
Citation Context ... W (µ) s ∫ ∞ = Ws − µs and B (ν) s 0 exp(B (ν) s )dW (µ) s , (22) = Bs − νs are two independent Brownian motions with respective drifts −µ ∈ R and −ν < 0, appear both in risk theory (see e.g. Paulsen =-=[55]-=-, Example 3.1) as well as in connection with invariant diffusions on the hyperbolic half-plane H (see Bougerol [16], Ikeda and Matsumoto [40], Baldi et al. [1], ...). Hence, it is no surprise that the... |

15 |
Poissonian exponential functionals, q-series, q-integrals, and the moment problem for log-normal distributions
- Bertoin, Biane, et al.
- 2004
(Show Context)
Citation Context ...= j=0 Γq(x) = ∞∏ (1 − aq j ), j=0 (q; q)∞ (qx (1 − q) ; q)∞ 1−x . We may now state the main formulas for the distribution of the exponential functional of the Poisson process, which are excerpts from =-=[10]-=- (see also [25]): The Laplace transform of I (q) ∞ is given by ( E exp(λI (q) ) 1 ∞ ) = (λ < 1), (13) (λ; q)∞ its Mellin transform by (( E I (q) ∞ ) s) = Γ(1 + s) Γq(1 + s)(1 − q) s = Γ(1 + s)(q1+s ; ... |

15 |
A diffusion process in a Brownian environment with drift
- Kawazu, Tanaka
- 1997
(Show Context)
Citation Context ... , is: ( ) 1 d −V (x) d exp(V (x)) e . 2 dx dx It is now clear, from Feller’s construction of such diffusions, that the potential V does not need to be differentiable. Brox [18] and Kawazu and Tanaka =-=[42, 43]-=- (see also e.g. Hu et al. [39] and Comtet et al. [23]) have studied this random diffusion in the case when V is a Brownian motion with negative drift : V (x) = Bx −kx, for k ≥ 0. Thus, it is no surpri... |

15 |
Krein condition in probabilistic moment problems
- Stoyanov
(Show Context)
Citation Context ...is a Poisson process, and more precisely, that the negative entire moments of the exponential functional of a Poisson process are closely related to those of the log-normal distribution). We refer to =-=[62, 64]-=- and the references therein for more about indeterminacy for the moment problem. 4. On the distribution of the exponential functional 4.1. Some important special cases In this section, we briefly pres... |

14 | Asymptotic laws for nonconservative self-similar fragmentations, Electron
- Bertoin, Gnedin
(Show Context)
Citation Context ...tion process. In particular, this approach yields important limit theorems for the empirical distribution of the fragments of self-similar fragmentations with a positive index of self-similarity; see =-=[9, 12, 17, 30]-=-. On the other hand, self-similar fragmentations with a negative index of self-similarity are dissipative, in the sense that the total mass of the fragments decreases as time passes. In this direction... |

13 | Rates of convergence of diffusions with drifted Brownian potentials
- Hu, Shi, et al.
- 1999
(Show Context)
Citation Context ... e . 2 dx dx It is now clear, from Feller’s construction of such diffusions, that the potential V does not need to be differentiable. Brox [18] and Kawazu and Tanaka [42, 43] (see also e.g. Hu et al. =-=[39]-=- and Comtet et al. [23]) have studied this random diffusion in the case when V is a Brownian motion with negative drift : V (x) = Bx −kx, for k ≥ 0. Thus, it is no surprise that the knowledge about th... |

13 | Self-similar processes with independent increments associated with Levy and Bessel processes
- Jeanblanc, Pitman, et al.
- 2002
(Show Context)
Citation Context ...ments are independent but not necessarily homogeneous. This situation arises naturally in particular in the representation of self-decomposable laws on R+ (see Sato [59] and Jeanblanc, Pitman and Yor =-=[41]-=-). It also occurs in in non-parametric Bayesian statistics, where the mean of a random distribution chosen from a neutral-to-the-right prior can be represented as the exponential functional of an incr... |

12 |
Present value distributions with applications to ruin theory and stochastic equations
- Gjessing, Paulsen
- 1997
(Show Context)
Citation Context ...ght-hand side, we recognize the k-th moment of a beta variable with parameter (1 − a, a + b − 1). We thus get the identity in law I∞ L = 1 β1−a,a+b−1 which has been discovered by Gjessing and Paulsen =-=[32]-=-. , 4.2. An integro-differential equation In this section, we present an equation satisfied by the density of the law of the exponential functional. We shall assume here that the Lévy process is given... |

11 |
Subordinators : examples and applications. Ecole d’Eté de Saint-Flour, LNM 1717
- Bertoin
- 1997
(Show Context)
Citation Context ...tic continuation of the characteristic exponent Ψ to the upper half-plane, namely Φ(q) = Ψ(iq), q ≥ 0 . The Lévy-Khintchine formula now reads ∫ Φ(q) = dq + (1 − e −qx )Π(dx), q ≥ 0. ]0,∞[ We refer to =-=[7]-=- for background on subordinators. The first part of the following theorem has been established by Carmona et al. [20], the second is an immediate extension of [15]. Theorem 2 Suppose that ξ is a subor... |

11 |
On the distribution of the sizes of particles which undergo splitting. Theory Probab
- Filippov
- 1961
(Show Context)
Citation Context ...tion process. In particular, this approach yields important limit theorems for the empirical distribution of the fragments of self-similar fragmentations with a positive index of self-similarity; see =-=[9, 12, 17, 30]-=-. On the other hand, self-similar fragmentations with a negative index of self-similarity are dissipative, in the sense that the total mass of the fragments decreases as time passes. In this direction... |

10 | A transformation from hausdorff to stieltjes moment sequences. Arkiv för matematik
- Berg, Duran
- 2004
(Show Context)
Citation Context ... (q + Φ(1)) · · · (q + Φ(k)) .J. Bertoin and M. Yor/Exponential functionals 196 The rest of the proof is immediate. (ii) We refer to [15] for the proof of the existence of ρ; see also Berg and Duran =-=[4]-=- for an analytical proof. The second part of the statement now follows from (i), since k! is the k-th moment of the standard exponential law and the latter is determined by its entire moments. In Sect... |

10 |
Regularity of the Half-Line for Lévy Processes
- Bertoin
- 1997
(Show Context)
Citation Context ...lows from the 0-1 law for ξ. The equivalence (iii) ⇔ (v) is the criterion of Rogozin (see Theorem VI.12 in [5]), whereas (iii) ⇔ (iv) ⇔ (vi) is a version of Erickson’s theorem for Lévy processes (see =-=[6]-=-). Finally, it is known (see Theorem VI.12 in [5]) that when (iii) fails, ξ visits the negative half-line at arbitrarily large times a.s. It is easily seen from an application of the strong Markov pro... |

10 |
Exemples de théorèmes locaux sur les groupes résolubles
- Bougerol
- 1983
(Show Context)
Citation Context ...th respective drifts −µ ∈ R and −ν < 0, appear both in risk theory (see e.g. Paulsen [55], Example 3.1) as well as in connection with invariant diffusions on the hyperbolic half-plane H (see Bougerol =-=[16]-=-, Ikeda and Matsumoto [40], Baldi et al. [1], ...). Hence, it is no surprise that the distribution of (22) has been much studied; it has a density given by exp(−2µ arctanx) f(x) = cµ,ν (1 + x2 ) ν+1/2... |

10 |
Exponential functionals of Brownian motion and disordered systems
- Comtet, Monthus, et al.
- 1996
(Show Context)
Citation Context ...clear, from Feller’s construction of such diffusions, that the potential V does not need to be differentiable. Brox [18] and Kawazu and Tanaka [42, 43] (see also e.g. Hu et al. [39] and Comtet et al. =-=[23]-=-) have studied this random diffusion in the case when V is a Brownian motion with negative drift : V (x) = Bx −kx, for k ≥ 0. Thus, it is no surprise that the knowledge about the exponential functiona... |

10 |
Brownian motion on the Hyperbolic plane and Selberg trace formula
- Ikeda, Matsumoto
- 1999
(Show Context)
Citation Context ...R and −ν < 0, appear both in risk theory (see e.g. Paulsen [55], Example 3.1) as well as in connection with invariant diffusions on the hyperbolic half-plane H (see Bougerol [16], Ikeda and Matsumoto =-=[40]-=-, Baldi et al. [1], ...). Hence, it is no surprise that the distribution of (22) has been much studied; it has a density given by exp(−2µ arctanx) f(x) = cµ,ν (1 + x2 ) ν+1/2 which belongs to the type... |

10 |
On the maximun of a diffusion process in a drifted brownian environment
- Kawazu, Tanaka
- 1993
(Show Context)
Citation Context ... , is: ( ) 1 d −V (x) d exp(V (x)) e . 2 dx dx It is now clear, from Feller’s construction of such diffusions, that the potential V does not need to be differentiable. Brox [18] and Kawazu and Tanaka =-=[42, 43]-=- (see also e.g. Hu et al. [39] and Comtet et al. [23]) have studied this random diffusion in the case when V is a Brownian motion with negative drift : V (x) = Bx −kx, for k ≥ 0. Thus, it is no surpri... |

8 | On powers of Stieltjes moment sequences - Berg |

8 |
A Markovian analysis of additiveincrease multiplicative-decrease algorithms
- Robert
(Show Context)
Citation Context ...∏ (1 − aq j ), j=0 (q; q)∞ (qx (1 − q) ; q)∞ 1−x . We may now state the main formulas for the distribution of the exponential functional of the Poisson process, which are excerpts from [10] (see also =-=[25]-=-): The Laplace transform of I (q) ∞ is given by ( E exp(λI (q) ) 1 ∞ ) = (λ < 1), (13) (λ; q)∞ its Mellin transform by (( E I (q) ∞ ) s) = Γ(1 + s) Γq(1 + s)(1 − q) s = Γ(1 + s)(q1+s ; q)∞ , (14) (q; ... |

8 |
Regularity of formation of dust in self-similar fragmentations
- HAAS
(Show Context)
Citation Context ...process, i.e. the time at which the mass of the tagged fragment vanishes. Distributional properties of the exponential functional provide a crucial tool for the analysis of the loss of mass; see Haas =-=[37, 38]-=-. In a related area, we mention that the terminal value of the exponential functional of subordinators also arises in the study of regenerative compositions, see the recent work of Gnedin, Pitman and ... |