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Tail asymptotics for exponential functionals of Lévy processes: the convolution equivalent case. (English. French summary) Zbl 1266.60086
Author’s 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 path-wise 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.

##### MSC:
 60G51 Processes with independent increments; Lévy processes 60E99 Distribution theory
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##### References:
 [1] J. Bertoin. Lévy Processes . Cambridge Tracts in Mathematics 121 . Cambridge Univ. Press, Cambridge, 1996. · Zbl 0861.60003 [2] J. Bertoin and R. A. Doney. Cramér’s estimate for Lévy processes. Statist. Probab. Lett. 21 (5) (1994) 363-365. · Zbl 0809.60085 [3] J. Bertoin and M. Yor. Exponential functionals of Lévy processes. Probab. Surv. 2 (2005) 191-212 (electronic). · Zbl 1189.60096 [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 0667.26003 [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 1191.60047 [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 0905.60056 [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. · Zbl 1241.60019 [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. [9] R. A. Doney and R. A. Maller. Cramér’s estimate for a reflected Lévy process. Ann. Appl. Probab. 15 (2005) 1445-1450. · Zbl 1069.60045 [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. · Zbl 0459.60017 [11] D. R. Grey. Regular variation in the tail behaviour of solutions of random difference equations. Ann. Appl. Probab. 4 (1) (1994) 169-183. · Zbl 0802.60057 [12] B. Haas. Loss of mass in deterministic and random fragmentations. Stochastic Process. Appl. 106 (2) (2003) 245-277. · Zbl 1075.60553 [13] H. Hult and F. Lindskog. Extremal behavior of stochastic integrals driven by regularly varying Lévy processes. Ann. Probab. 35 (1) (2007) 309-339. · Zbl 1121.60029 [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. · Zbl 1066.60049 [15] A. E. Kyprianou. Introductory Lectures on Fluctuations of Lévy Processes with Applications. Universitext . Springer, Berlin, 2006. · Zbl 1104.60001 [16] K. Maulik and B. Zwart. Tail asymptotics for exponential functionals of Lévy processes. Stochastic Process. Appl. 116 (2) (2006) 156-177. · Zbl 1090.60046 [17] A. G. Pakes. Convolution equivalence and infinite divisibility. J. Appl. Probab. 41 (2) (2004) 407-424. · Zbl 1051.60019 [18] A. G. Pakes. Convolution equivalence and infinite divisibility: Corrections and corollaries. J. Appl. Probab. 44 (2) (2007) 295-305. · Zbl 1132.60015 [19] J. C. Pardo. On the future infimum of positive self-similar Markov processes. Stochastics 78 (3) (2006) 123-155. · Zbl 1100.60018 [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. · Zbl 1180.31010 [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. · Zbl 1162.60015 [22] P. Patie. Law of the absorption time of some positive self-similar Markov processes. Ann. Probab. 40 (2) (2012) 765-787. · Zbl 1241.60020 [23] V. Rivero. A law of iterated logarithm for increasing self-similar Markov processes. Stoch. Stoch. Rep. 75 (6) (2003) 443-472. · Zbl 1053.60027 [24] V. Rivero. Recurrent extensions of self-similar Markov processes and Cramér’s condition. Bernoulli 11 (3) (2005) 471-509. · Zbl 1077.60055 [25] V. Rivero. Recurrent extensions of self-similar Markov processes and Cramér’s condition II. Bernoulli 13 (2007) 1053-1070. · Zbl 1132.60056 [26] V. Rivero. Sinaĭ’s condition for real valued Lévy processes. Ann. Inst. Henri Poincaré Probab. Stat. 43 (3) (2007) 299-319. · Zbl 1115.60049 [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. · Zbl 0532.60065 [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. · Zbl 0533.60021 [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. · Zbl 1146.60014 [31] S. J. Wolfe. On a continuous analogue of the stochastic difference equation $$X_{n}=\rho X_{n-1}+B_{n}$$. Stochastic Process. Appl. 12 (3) (1982) 301-312. · Zbl 0482.60062
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