Tail asymptotics for exponential functionals of Lévy processes: The convolution equivalent case

Víctor Rivero

Annales de l'I.H.P. Probabilités et statistiques (2012)

  • Volume: 48, Issue: 4, page 1081-1102
  • ISSN: 0246-0203

Abstract

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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.

How to cite

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Rivero, Víctor. "Tail asymptotics for exponential functionals of Lévy processes: The convolution equivalent case." Annales de l'I.H.P. Probabilités et statistiques 48.4 (2012): 1081-1102. <http://eudml.org/doc/271989>.

@article{Rivero2012,
abstract = {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.},
author = {Rivero, Víctor},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {convolution equivalent distributions; exponential functionals of Lévy processes; fluctuation theory of Lévy processes; exponential functionals of Levy processes; fluctuation theory of Levy processes},
language = {eng},
number = {4},
pages = {1081-1102},
publisher = {Gauthier-Villars},
title = {Tail asymptotics for exponential functionals of Lévy processes: The convolution equivalent case},
url = {http://eudml.org/doc/271989},
volume = {48},
year = {2012},
}

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
PB - Gauthier-Villars
VL - 48
IS - 4
SP - 1081
EP - 1102
AB - 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.
LA - eng
KW - convolution equivalent distributions; exponential functionals of Lévy processes; fluctuation theory of Lévy processes; exponential functionals of Levy processes; fluctuation theory of Levy processes
UR - http://eudml.org/doc/271989
ER -

References

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  1. [1] J. Bertoin. Lévy Processes. Cambridge Tracts in Mathematics 121. Cambridge Univ. Press, Cambridge, 1996. Zbl0861.60003
  2. [2] J. Bertoin and R. A. Doney. Cramér’s estimate for Lévy processes. Statist. Probab. Lett. 21 (5) (1994) 363–365. Zbl0809.60085
  3. [3] J. Bertoin and M. Yor. Exponential functionals of Lévy processes. Probab. Surv. 2 (2005) 191–212 (electronic). Zbl1189.60096
  4. [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. Zbl0667.26003
  5. [5] M.-E. Caballero and V. Rivero. On the asymptotic behaviour of increasing self-similar Markov processes. Electron. J. Probab.14 (2009) 865–894. Zbl1191.60047
  6. [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. Zbl0905.60056
  7. [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. Zbl1241.60019MR2917773
  8. [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. Zbl1128.60036MR2320889
  9. [9] R. A. Doney and R. A. Maller. Cramér’s estimate for a reflected Lévy process. Ann. Appl. Probab.15 (2005) 1445–1450. Zbl1069.60045MR2134110
  10. [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. Zbl0459.60017MR614631
  11. [11] D. R. Grey. Regular variation in the tail behaviour of solutions of random difference equations. Ann. Appl. Probab. 4 (1) (1994) 169–183. Zbl0802.60057MR1258178
  12. [12] B. Haas. Loss of mass in deterministic and random fragmentations. Stochastic Process. Appl. 106 (2) (2003) 245–277. Zbl1075.60553MR1989629
  13. [13] H. Hult and F. Lindskog. Extremal behavior of stochastic integrals driven by regularly varying Lévy processes. Ann. Probab. 35 (1) (2007) 309–339. Zbl1121.60029MR2303951
  14. [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. Zbl1066.60049MR2099651
  15. [15] A. E. Kyprianou. Introductory Lectures on Fluctuations of Lévy Processes with Applications. Universitext. Springer, Berlin, 2006. Zbl1104.60001MR2250061
  16. [16] K. Maulik and B. Zwart. Tail asymptotics for exponential functionals of Lévy processes. Stochastic Process. Appl. 116 (2) (2006) 156–177. Zbl1090.60046MR2197972
  17. [17] A. G. Pakes. Convolution equivalence and infinite divisibility. J. Appl. Probab. 41 (2) (2004) 407–424. Zbl1051.60019MR2052581
  18. [18] A. G. Pakes. Convolution equivalence and infinite divisibility: Corrections and corollaries. J. Appl. Probab. 44 (2) (2007) 295–305. Zbl1132.60015MR2340199
  19. [19] J. C. Pardo. On the future infimum of positive self-similar Markov processes. Stochastics 78 (3) (2006) 123–155. Zbl1100.60018MR2241913
  20. [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. Zbl1180.31010MR2548498
  21. [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. Zbl1162.60015MR2537239
  22. [22] P. Patie. Law of the absorption time of some positive self-similar Markov processes. Ann. Probab. 40 (2) (2012) 765–787. Zbl1241.60020MR2952091
  23. [23] V. Rivero. A law of iterated logarithm for increasing self-similar Markov processes. Stoch. Stoch. Rep. 75 (6) (2003) 443–472. Zbl1053.60027MR2029617
  24. [24] V. Rivero. Recurrent extensions of self-similar Markov processes and Cramér’s condition. Bernoulli 11 (3) (2005) 471–509. Zbl1077.60055MR2146891
  25. [25] V. Rivero. Recurrent extensions of self-similar Markov processes and Cramér’s condition II. Bernoulli13 (2007) 1053–1070. Zbl1132.60056MR2364226
  26. [26] V. Rivero. Sina’s condition for real valued Lévy processes. Ann. Inst. Henri Poincaré Probab. Stat. 43 (3) (2007) 299–319. Zbl1115.60049MR2319699
  27. [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. Zbl0532.60065MR736019
  28. [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. Zbl0533.60021MR738769
  29. [29] V. Vigon. Simplifiez vos Lévy en titillant la factorisation de Wiener–Hopf. Thèse de doctorat de l’INSA de Rouen, 2002. 
  30. [30] T. Watanabe. Convolution equivalence and distributions of random sums. Probab. Theory Related Fields 142 (3–4) (2008) 367–397. Zbl1146.60014MR2438696
  31. [31] S. J. Wolfe. On a continuous analogue of the stochastic difference equation X n = ρ X n - 1 + B n . Stochastic Process. Appl. 12 (3) (1982) 301–312. Zbl0482.60062MR656279

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