Tail asymptotics for exponential functionals of Lévy processes: The convolution equivalent case
Annales de l'I.H.P. Probabilités et statistiques, Volume 48 (2012) no. 4, pp. 1081-1102.

We determine the rate of decrease of the right tail distribution of the exponential functional of a Lévy process with a convolution equivalent Lévy measure. Our main result establishes that it decreases as the right tail of the image under the exponential function of the Lévy measure of the underlying Lévy process. The method of proof relies on fluctuation theory of Lévy processes and an explicit pathwise representation of the exponential functional as the exponential functional of a bivariate subordinator. Our techniques allow us to establish analogous results under the excursion measure of the underlying Lévy process reflected in its past infimum.

On s'intéresse à la vitesse de décroissance de la queue de distribution d'une fonctionnelle exponentielle d'un processus de Lévy dont la mesure de sauts est équivalente par convolution. Le résultat principal de ce papier montre que cette vitesse décroît comme la queue de la mesure image de la mesure de sauts par la fonction exponentielle. La preuve de ce résultat repose sur la théorie des fluctuations pour les processus de Lévy et une représentation trajectorielle explicite de la fonctionnelle exponentielle comme la fonctionnelle exponentielle d'un subordinateur bivarié. Nos techniques nous permettent également d'établir des résultats similaires sous la mesure d'excursion du processus de Lévy sous-jacent réfléchi en son minimum passé.

DOI: 10.1214/12-AIHP477
Classification: 60G51(60F99)
Keywords: convolution equivalent distributions, exponential functionals of Lévy processes, fluctuation theory of Lévy processes
@article{AIHPB_2012__48_4_1081_0,
     author = {Rivero, V{\'\i}ctor},
     title = {Tail asymptotics for exponential functionals of {L\'evy} processes: {The} convolution equivalent case},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {1081--1102},
     publisher = {Gauthier-Villars},
     volume = {48},
     number = {4},
     year = {2012},
     doi = {10.1214/12-AIHP477},
     mrnumber = {3052404},
     zbl = {1266.60086},
     language = {en},
     url = {http://www.numdam.org/articles/10.1214/12-AIHP477/}
}
TY  - JOUR
AU  - Rivero, Víctor
TI  - Tail asymptotics for exponential functionals of Lévy processes: The convolution equivalent case
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2012
SP  - 1081
EP  - 1102
VL  - 48
IS  - 4
PB  - Gauthier-Villars
UR  - http://www.numdam.org/articles/10.1214/12-AIHP477/
DO  - 10.1214/12-AIHP477
LA  - en
ID  - AIHPB_2012__48_4_1081_0
ER  - 
%0 Journal Article
%A Rivero, Víctor
%T Tail asymptotics for exponential functionals of Lévy processes: The convolution equivalent case
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2012
%P 1081-1102
%V 48
%N 4
%I Gauthier-Villars
%U http://www.numdam.org/articles/10.1214/12-AIHP477/
%R 10.1214/12-AIHP477
%G en
%F AIHPB_2012__48_4_1081_0
Rivero, Víctor. Tail asymptotics for exponential functionals of Lévy processes: The convolution equivalent case. Annales de l'I.H.P. Probabilités et statistiques, Volume 48 (2012) no. 4, pp. 1081-1102. doi : 10.1214/12-AIHP477. http://www.numdam.org/articles/10.1214/12-AIHP477/

[1] J. Bertoin. Lévy Processes. Cambridge Tracts in Mathematics 121. Cambridge Univ. Press, Cambridge, 1996. | Zbl

[2] J. Bertoin and R. A. Doney. Cramér's estimate for Lévy processes. Statist. Probab. Lett. 21 (5) (1994) 363-365. | Zbl

[3] J. Bertoin and M. Yor. Exponential functionals of Lévy processes. Probab. Surv. 2 (2005) 191-212 (electronic). | Zbl

[4] N. H. Bingham, C. M. Goldie and J. L. Teugels. Regular Variation. Encyclopedia of Mathematics and Its Applications 27. Cambridge Univ. Press, Cambridge, 1989. | Zbl

[5] M.-E. Caballero and V. Rivero. On the asymptotic behaviour of increasing self-similar Markov processes. Electron. J. Probab. 14 (2009) 865-894. | Zbl

[6] P. Carmona, F. Petit and M. Yor. On the distribution and asymptotic results for exponential functionals of Lévy processes. In Exponential Functionals and Principal Values Related to Brownian Motion 73-130. Rev. Mat. Iberoamericana, Madrid, 1997. | Zbl

[7] L. Chaumont, A. E. Kyprianou, J. C. Pardo and V. Rivero. Fluctuation theory and exit systems for positive self-similar Markov processes. Ann. Probab. 40 (1) (2012) 245-279. | MR | Zbl

[8] R. A. Doney. Fluctuation Theory for Lévy Processes. Lectures from the 35th Summer School on Probability Theory Held in Saint-Flour, July 6-23, 2005. Lecture Notes in Mathematics 1897. Springer, Berlin, 2007. | MR | Zbl

[9] R. A. Doney and R. A. Maller. Cramér's estimate for a reflected Lévy process. Ann. Appl. Probab. 15 (2005) 1445-1450. | MR | Zbl

[10] P. Embrechts and C. M. Goldie. Comparing the tail of an infinitely divisible distribution with integrals of its Lévy measure. Ann. Probab. 9 (3) (1981) 468-481. | MR | Zbl

[11] D. R. Grey. Regular variation in the tail behaviour of solutions of random difference equations. Ann. Appl. Probab. 4 (1) (1994) 169-183. | MR | Zbl

[12] B. Haas. Loss of mass in deterministic and random fragmentations. Stochastic Process. Appl. 106 (2) (2003) 245-277. | MR | Zbl

[13] H. Hult and F. Lindskog. Extremal behavior of stochastic integrals driven by regularly varying Lévy processes. Ann. Probab. 35 (1) (2007) 309-339. | MR | Zbl

[14] C. Klüppelberg, A. E. Kyprianou and R. A. Maller. Ruin probabilities and overshoots for general Lévy insurance risk processes. Ann. Appl. Probab. 14 (4) (2004) 1766-1801. | MR | Zbl

[15] A. E. Kyprianou. Introductory Lectures on Fluctuations of Lévy Processes with Applications. Universitext. Springer, Berlin, 2006. | MR | Zbl

[16] K. Maulik and B. Zwart. Tail asymptotics for exponential functionals of Lévy processes. Stochastic Process. Appl. 116 (2) (2006) 156-177. | MR | Zbl

[17] A. G. Pakes. Convolution equivalence and infinite divisibility. J. Appl. Probab. 41 (2) (2004) 407-424. | MR | Zbl

[18] A. G. Pakes. Convolution equivalence and infinite divisibility: Corrections and corollaries. J. Appl. Probab. 44 (2) (2007) 295-305. | MR | Zbl

[19] J. C. Pardo. On the future infimum of positive self-similar Markov processes. Stochastics 78 (3) (2006) 123-155. | MR | Zbl

[20] P. Patie. Infinite divisibility of solutions to some self-similar integro-differential equations and exponential functionals of Lévy processes. Ann. Inst. Henri Poincaré Probab. Stat. 45 (3) (2009) 667-684. | EuDML | Numdam | MR | Zbl

[21] P. Patie. Law of the exponential functional of one-sided Lévy processes and Asian options. C. R. Math. Acad. Sci. Paris 347 (7-8) (2009) 407-411. | MR | Zbl

[22] P. Patie. Law of the absorption time of some positive self-similar Markov processes. Ann. Probab. 40 (2) (2012) 765-787. | MR | Zbl

[23] V. Rivero. A law of iterated logarithm for increasing self-similar Markov processes. Stoch. Stoch. Rep. 75 (6) (2003) 443-472. | MR | Zbl

[24] V. Rivero. Recurrent extensions of self-similar Markov processes and Cramér's condition. Bernoulli 11 (3) (2005) 471-509. | MR | Zbl

[25] V. Rivero. Recurrent extensions of self-similar Markov processes and Cramér's condition II. Bernoulli 13 (2007) 1053-1070. | MR | Zbl

[26] V. Rivero. Sina | Numdam | MR | Zbl

[27] K.-I. Sato and M. Yamazato. Stationary processes of Ornstein-Uhlenbeck type. In Probability Theory and Mathematical Statistics (Tbilisi, 1982) 541-551. Lecture Notes in Math. 1021. Springer, Berlin, 1983. | MR | Zbl

[28] K.-I. Sato and M. Yamazato. Operator-self-decomposable distributions as limit distributions of processes of Ornstein-Uhlenbeck type. Stochastic Process. Appl. 17 (1) (1984) 73-100. | MR | Zbl

[29] V. Vigon. Simplifiez vos Lévy en titillant la factorisation de Wiener-Hopf. Thèse de doctorat de l'INSA de Rouen, 2002.

[30] T. Watanabe. Convolution equivalence and distributions of random sums. Probab. Theory Related Fields 142 (3-4) (2008) 367-397. | MR | Zbl

[31] S. J. Wolfe. On a continuous analogue of the stochastic difference equation X n =ρX n-1 +B n . Stochastic Process. Appl. 12 (3) (1982) 301-312. | MR | Zbl

Cited by Sources: