## On The Distribution And Asymptotic Results For Exponential Functionals Of Lévy Processes (1997)

Citations: | 71 - 9 self |

### BibTeX

@MISC{Carmona97onthe,

author = {Philippe Carmona and Frédérique Petit and Erique Petit and Marc Yor},

title = {On The Distribution And Asymptotic Results For Exponential Functionals Of Lévy Processes},

year = {1997}

}

### Years of Citing Articles

### OpenURL

### Abstract

. 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 ff-stable, we prove that t \Gamma1=ff log A t converges in law, as t !1, to the supremum of an ff-stable 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...

### Citations

1055 |
Continuous Martingales and Brownian Motion. Third Edition
- Revuz, Yor
- 1999
(Show Context)
Citation Context ...m on the right hand side converges almost surely to sup s1ss . Remark 4.3. When ff = 2, i.e.sis a Brownian motion, the preceding result is well-known (see e.g. Durrett [12], Pitman-Yor [25], RevuzYor =-=[26]-=-, Chapter I, Exercise 1.18, or Bertoin-Werner [3] and [4]), and is a main step in the proof of Spitzer's theorem on the winding number of planar Brownian motion. It is natural to think that the assump... |

425 |
Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance, Chapman
- Samorodnitsky, Taqqu
- 1994
(Show Context)
Citation Context ... a random L'evy environment. Since the proof of Proposition 2.1 relies heavily on the properties of generalized Ornstein-Uhlenbeck processes (see Hadjiev[18], Novikov [24] and Samorodnitsky and Taqqu =-=[27]-=-, section 3.6), we give in Appendix 1 a summary of the properties of these processes. Eventually, we devote Appendix 2 to the study of a Poisson process with drift : there, we apply the results of Sec... |

415 |
Special Functions and their Applications
- Lebedev
- 1965
(Show Context)
Citation Context ...Gamma(b + iu) \Gamma(b) ; so that : OE (b) (u) = log ` \Gamma(b + iu) \Gamma(b) ' : Then, the representation of OE (b) derives from the formulae, valid for Re(z) ? 0, (see Formula (1.3.14) of Lebedev =-=[22]-=-) : log \Gamma(z) = Z z 1 /(x) dx ; /(z) = \Gammafl + Z 1 0 e \Gammat \Gamma e \Gammatz 1 \Gamma e \Gammat dt : (iv) Holder's inequality implies that the function g() = 1 1\Gammaff log E \Theta H ff i... |

187 |
Random difference equations and renewal theory for products of random matrices
- Kesten
- 1973
(Show Context)
Citation Context ...1. The stochastic (difference) equation. The stochastic difference equation Y n = A n Y n\Gamma1 +B n with f(A n ; B n ); n 2 Ng an i.i.d sequence, has been thoroughly studied by Vervaat [29], Kesten =-=[20]-=-, and more recently by Brandt [8]. This equation arises in various disciplines, for example economics, physics, nuclear technology, ... (see Vervaat [29]) If we have the convergence Y n d 7\Gamma! n!1... |

113 | Exponential Functionals of Brownian Motion and Related Processes
- Yor
- 2001
(Show Context)
Citation Context ...s ds whensis a Brownian motion with drift, and t may be finite or infinite. This has been studied by a number of authors, including Bouchaud-Sornette [6], Dufresne [11], Geman-Yor [15] and Yor ([31], =-=[30]-=-). c) In her PhD Thesis [23], C'ecile Monthus shows how the exponential functional naturally arises in the study of a one-dimensional diffusion in a frozen random potential V . Let us give examples of... |

94 |
On a stochastic difference equation and a representation of non-negative infinitely divisible random variables
- Vervaat
- 1979
(Show Context)
Citation Context ...s pricing. 6.1. The stochastic (difference) equation. The stochastic difference equation Y n = A n Y n\Gamma1 +B n with f(A n ; B n ); n 2 Ng an i.i.d sequence, has been thoroughly studied by Vervaat =-=[29]-=-, Kesten [20], and more recently by Brandt [8]. This equation arises in various disciplines, for example economics, physics, nuclear technology, ... (see Vervaat [29]) If we have the convergence Y n d... |

92 |
The distribution of a perpetuity, with application to risk theory and pension funding
- Dufresne
- 1990
(Show Context)
Citation Context ...he distribution of A t () = R t 0 e s ds whensis a Brownian motion with drift, and t may be finite or infinite. This has been studied by a number of authors, including Bouchaud-Sornette [6], Dufresne =-=[11]-=-, Geman-Yor [15] and Yor ([31], [30]). c) In her PhD Thesis [23], C'ecile Monthus shows how the exponential functional naturally arises in the study of a one-dimensional diffusion in a frozen random p... |

89 |
Probabilites et potentiel
- Dellacherie, Meyer
- 1987
(Show Context)
Citation Context ...-stopping time T: E [jM1 \Gamma M T \Gamma j=F T ]s2 OE(1) e \Gamma + T \Gamma 2 OE(1) : Hence, the martingale (M t ; ts0) is in BMO, and kMkBMO 1s2 OE(1) . Thanks to John-Nirenberg's inequality (see =-=[10]-=-, Chapter VI, Theorem 109), if 0sff ! 1 2OE(1) , we have: E \Theta e ffA1 ! +1 : In fact, we obtain better results : Proposition 3.3. The law of A1 is determined by its integral moments m n := E [A n ... |

78 |
Teugels. Regular variation, volume 27 of Encyclopedia of Mathematics and its Applications
- Bingham, Goldie, et al.
- 1989
(Show Context)
Citation Context ... us2t : and : sup t0 1 t E [M 2 t ]s2. Now, remark that the equivalent (4.6) is an immediate consequence of the convergence (4.5) applied to the process \Gamma, the Monotone Density Theorem (see e.g. =-=[5]-=-, Theorem 1.7.2), and the relation : d dt E [log A t (\Gamma)] = E e \Gamma t A t (\Gamma) = E 1 A t () ; (the last equality coming from Lemma 2.3). We now show how to use Girsanov transform to estima... |

75 |
The stochastic equation Yn+1 = AnYn + Bn with stationary coefcients
- Brandt
- 1986
(Show Context)
Citation Context ...ation. The stochastic difference equation Y n = A n Y n\Gamma1 +B n with f(A n ; B n ); n 2 Ng an i.i.d sequence, has been thoroughly studied by Vervaat [29], Kesten [20], and more recently by Brandt =-=[8]-=-. This equation arises in various disciplines, for example economics, physics, nuclear technology, ... (see Vervaat [29]) If we have the convergence Y n d 7\Gamma! n!1 Y , then Y satisfies the stochas... |

69 |
Bessel processes, asian options, and perpetuities, Mathematical Finance 3
- Geman, Yor
- 1993
(Show Context)
Citation Context ...of A t () = R t 0 e s ds whensis a Brownian motion with drift, and t may be finite or infinite. This has been studied by a number of authors, including Bouchaud-Sornette [6], Dufresne [11], Geman-Yor =-=[15]-=- and Yor ([31], [30]). c) In her PhD Thesis [23], C'ecile Monthus shows how the exponential functional naturally arises in the study of a one-dimensional diffusion in a frozen random potential V . Let... |

63 |
Semi-stable Markov processes
- Lamperti
- 1972
(Show Context)
Citation Context ...t ; ts0 ; P x ; x 2 E), with state space E = R+ ; R or R n , such that : ( 1 a ae X at ; ts0 ; P a ae x ) d = (X t ; ts0 ; P x ) (a ? 0; x 2 E): Such Markov processes have been considered by Lamperti =-=[21]-=- in relation with exponentials of L'evy processes. Let ( t ; ts0 ) be a L'evy process, starting from zero, independent from X, and let A t = A t (\Gamma) = R t 0 e \Gamma s ds. Proposition 5.5. The ge... |

60 |
The Gamma Function
- Artin
- 1964
(Show Context)
Citation Context ...ammat dt : (iv) Holder's inequality implies that the function g() = 1 1\Gammaff log E \Theta H ff is convex on R+ . Since g(0) = 0 and g( + 1) = g() + log( + 1), we conclude from Theorem 2.1 of Artin =-=[2]-=- (see also Artin [1] or Bourbaki [7], Chapter 7 Proposition 1 and exercises 1 and 2), that fors? 0, g() = log \Gamma(+1). Hence, by analytic continuation : E \Theta H iu ff = \Gamma(1 + iu) 1\Gammaff ... |

33 |
Some aspects of Brownian motion. Part I. Some special functionals
- Yor
- 1992
(Show Context)
Citation Context ...t 0 e s ds whensis a Brownian motion with drift, and t may be finite or infinite. This has been studied by a number of authors, including Bouchaud-Sornette [6], Dufresne [11], Geman-Yor [15] and Yor (=-=[31]-=-, [30]). c) In her PhD Thesis [23], C'ecile Monthus shows how the exponential functional naturally arises in the study of a one-dimensional diffusion in a frozen random potential V . Let us give examp... |

25 |
The Black-Scholes option pricing problem in mathematical finance: generalization and extensions for a large class of stochastic processes
- Bouchaud, Sornette
- 1994
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Citation Context ...nterested in the distribution of A t () = R t 0 e s ds whensis a Brownian motion with drift, and t may be finite or infinite. This has been studied by a number of authors, including Bouchaud-Sornette =-=[6]-=-, Dufresne [11], Geman-Yor [15] and Yor ([31], [30]). c) In her PhD Thesis [23], C'ecile Monthus shows how the exponential functional naturally arises in the study of a one-dimensional diffusion in a ... |

20 |
On the Azéma martingales
- Émery
- 1990
(Show Context)
Citation Context ...lly, we devote Appendix 2 to the study of a Poisson process with drift : there, we apply the results of Sections 2, 3, 4, and make explicit the connection with the Az'ema-Emery martingales (see Emery =-=[13]-=-). Contents 1. Introduction 1 2. An integro-differential equation satisfied by the density of the exponential functional 4 2.1. Proof of Proposition 2.1. 7 3. The moments formulae 10 3.1. A functional... |

17 |
A stochastic approach to the gamma function
- Gordon
- 1994
(Show Context)
Citation Context ...) is a L'evy process such that for each a ? 0, Z a is a gamma variable of shape parameter a. Then, we consider (Z (j) ; j 2 N) a family of independent gamma processes (Z (j) a ; as0).Following Gordon =-=[17]-=-, we fix b ? 0 and define j (b) a := \Gammaafl + 1 X j=0 a j + 1 \Gamma Z (j) a j + b ; where fl = \Gamma\Gamma 0 (1) denotes the Euler constant. This definition makes sense, as we may write alternati... |

14 |
Exponential functionals of Brownian motion and disordered systems
- Comtet, Monthus, et al.
- 1998
(Show Context)
Citation Context ...ntial functional naturally arises in the study of a one-dimensional diffusion in a frozen random potential V . Let us give examples of physical quantities considered in [23] and in Comtet-Monthus-Yor =-=[9]-=- : ffl The thermal-average time spent in the interval [x; x+dx] is T (x) dx, where T (x) = 2 Z +1 x e V (y) dy : ffl The stationary flow JN of particles going through a disordered sample of size N wit... |

14 |
Asymptotic laws of planar Brownian motion
- Pitman, Yor
- 1986
(Show Context)
Citation Context ... the second term on the right hand side converges almost surely to sup s1ss . Remark 4.3. When ff = 2, i.e.sis a Brownian motion, the preceding result is well-known (see e.g. Durrett [12], Pitman-Yor =-=[25]-=-, RevuzYor [26], Chapter I, Exercise 1.18, or Bertoin-Werner [3] and [4]), and is a main step in the proof of Spitzer's theorem on the winding number of planar Brownian motion. It is natural to think ... |

12 |
Fonctions d’une variable réelle
- Bourbaki
- 1965
(Show Context)
Citation Context ...implies that the function g() = 1 1\Gammaff log E \Theta H ff is convex on R+ . Since g(0) = 0 and g( + 1) = g() + log( + 1), we conclude from Theorem 2.1 of Artin [2] (see also Artin [1] or Bourbaki =-=[7]-=-, Chapter 7 Proposition 1 and exercises 1 and 2), that fors? 0, g() = log \Gamma(+1). Hence, by analytic continuation : E \Theta H iu ff = \Gamma(1 + iu) 1\Gammaff (u 2 R) ; which implies the desired ... |

11 |
On the maximum of a diffusion process in a drifted Brownian environment, Seminaire de probabilités XXVII
- Kawazu, Tanaka
- 1993
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Citation Context ...he 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 motion with drift. a) In =-=[19]-=-, Kawazu and Tanaka investigate the asymptotic behavior of the tail of the distribution of the maximum of a diffusion process in a drifted Brownian environment, precisely : given a Brownian motion (W ... |

8 |
On certain selfdecomposable distributions
- Shanbhag, Sreehari
- 1977
(Show Context)
Citation Context ... derivative of the Gamma function. (iv) For each ff 2 (0; 1), we have : Lsff = log H 1\Gammasff d = j (1)sff Remark 3.5. The fact that log Z a is infinitely divisible is well-known: see e.g. Shanbhag =-=[28]-=-, who shows that log Z a is self-decomposable, hence infinitely divisible. Proof. (i) Let a 0 ? 0. Since, for each j 2 N, (Z (j) a+a 0 \Gamma Z (j) a 0 ; as0) is a gamma process independent of ( Z (j)... |

7 |
The first passage problem for generalized Ornstein-Uhlenbeck processes with non-positive jumps
- Hadjiev
- 1985
(Show Context)
Citation Context ...y to the study of the maximum of a diffusion in a random L'evy environment. Since the proof of Proposition 2.1 relies heavily on the properties of generalized Ornstein-Uhlenbeck processes (see Hadjiev=-=[18]-=-, Novikov [24] and Samorodnitsky and Taqqu [27], section 3.6), we give in Appendix 1 a summary of the properties of these processes. Eventually, we devote Appendix 2 to the study of a Poisson process ... |

6 |
W.: Asymptotic windings of planar Brownian motion revisited via the Ornstein-Uhlenbeck process
- Bertoin, Werner
- 1994
(Show Context)
Citation Context ...o sup s1ss . Remark 4.3. When ff = 2, i.e.sis a Brownian motion, the preceding result is well-known (see e.g. Durrett [12], Pitman-Yor [25], RevuzYor [26], Chapter I, Exercise 1.18, or Bertoin-Werner =-=[3]-=- and [4]), and is a main step in the proof of Spitzer's theorem on the winding number of planar Brownian motion. It is natural to think that the assumptions of the preceding Proposition were too restr... |

5 |
Étude de quelques fonctionnelles du mouvement Brownien et de certaines propriétés de la diffusion unidimensionnelle en milieu aléatoire, Thèse de Doctorat de l’Université Paris VI
- Monthus
- 1995
(Show Context)
Citation Context ...on with drift, and t may be finite or infinite. This has been studied by a number of authors, including Bouchaud-Sornette [6], Dufresne [11], Geman-Yor [15] and Yor ([31], [30]). c) In her PhD Thesis =-=[23]-=-, C'ecile Monthus shows how the exponential functional naturally arises in the study of a one-dimensional diffusion in a frozen random potential V . Let us give examples of physical quantities conside... |

5 |
On the first passage time of an autoregressive process over a level and an application to a “disorder” problem. Theory Probab
- Novikov
- 1990
(Show Context)
Citation Context ... of the maximum of a diffusion in a random L'evy environment. Since the proof of Proposition 2.1 relies heavily on the properties of generalized Ornstein-Uhlenbeck processes (see Hadjiev[18], Novikov =-=[24]-=- and Samorodnitsky and Taqqu [27], section 3.6), we give in Appendix 1 a summary of the properties of these processes. Eventually, we devote Appendix 2 to the study of a Poisson process with drift : t... |

5 |
Random Brownian scaling and some absolute continuity relationships
- Yor
- 1992
(Show Context)
Citation Context ... the roots of : 0 = (x + a) 2 (x + b) 2 +s2 (x + b) 2 +s2 (x + a) 2 . (2) If X a := R 1 0 R s (4a) ds, where (R s (ffi); ss0) is the square of a ffidimensional Bessel process starting from 0 (see Yor =-=[32]-=-), then : E exp ` \Gamma 2 2 A1 (h a ) ' = c a E X \Gamma 1 2 a exp(\Gamma 2 2 X a ) : This equation provides another way of obtaining the density of X a . 6.4. The relationship with Az'ema-Emery mart... |

4 |
Einführung in die Theorie der Gammafunktion
- Artin
- 1931
(Show Context)
Citation Context ...er's inequality implies that the function g() = 1 1\Gammaff log E \Theta H ff is convex on R+ . Since g(0) = 0 and g( + 1) = g() + log( + 1), we conclude from Theorem 2.1 of Artin [2] (see also Artin =-=[1]-=- or Bourbaki [7], Chapter 7 Proposition 1 and exercises 1 and 2), that fors? 0, g() = log \Gamma(+1). Hence, by analytic continuation : E \Theta H iu ff = \Gamma(1 + iu) 1\Gammaff (u 2 R) ; which impl... |

2 |
The Theory of Stochastic Processes II, volume 218 of Die Grundlehren der mathematischen Wissenchaften in Einzeldarstellungen
- Gihman, Skorohod
- 1975
(Show Context)
Citation Context ...+s2 , such thats1 (resp.s2 ) has no positive (resp. negative) jumps. Since M ()sM ( 1 ) + M ( 2 ) we may suppose that the jumpssare of a fixed sign, and apply Theorem 8, Chapter IV of Gihman-Skorohod =-=[16]-=-, to conclude. Remark 3.2. For every positive measurable function f , we have : E [f(A T )] = ` Z 1 0 e \Gamma`t E [f(A t )] dt : Hence, the moments formula (1.2) characterizes the law of A t at fixed... |

1 |
windings. The Annals of Probability
- Stable
- 1996
(Show Context)
Citation Context ...ss . Remark 4.3. When ff = 2, i.e.sis a Brownian motion, the preceding result is well-known (see e.g. Durrett [12], Pitman-Yor [25], RevuzYor [26], Chapter I, Exercise 1.18, or Bertoin-Werner [3] and =-=[4]-=-), and is a main step in the proof of Spitzer's theorem on the winding number of planar Brownian motion. It is natural to think that the assumptions of the preceding Proposition were too restrictive, ... |

1 |
A new proof of Spitzer's result on the winding number of 2dimensional Brownian motion
- Durrett
- 1982
(Show Context)
Citation Context ...method shows that the second term on the right hand side converges almost surely to sup s1ss . Remark 4.3. When ff = 2, i.e.sis a Brownian motion, the preceding result is well-known (see e.g. Durrett =-=[12]-=-, Pitman-Yor [25], RevuzYor [26], Chapter I, Exercise 1.18, or Bertoin-Werner [3] and [4]), and is a main step in the proof of Spitzer's theorem on the winding number of planar Brownian motion. It is ... |