Tail asymptotics for exponential functionals of Lévy processes

Stochastic Processes and their Applications

Volumn 116, Issue 2, 2006, Pages 156-177

  Maulik, Krishanu ^a,d   Zwart, Bert ^b,c  

^a EURANDOM   (Netherlands)

^b EINDHOVEN UNIVERSITY OF TECHNOLOGY   (Netherlands)

^c CWI   (Netherlands)

^d INDIAN STATISTICAL INSTITUTE   (India)

Author keywords

Breiman's theorem; Mellin transforms; Perpetuities; Subexponential distributions; Tauberian theorems

Indexed keywords

ASYMPTOTIC STABILITY; FUNCTION EVALUATION; PARETO PRINCIPLE; RANDOM PROCESSES;

BREIMAN'S THEOREM; MELLIN TRANSFORMS; PERPETUITIES; SUBEXPONENTIAL DISTRIBUTIONS; TAUBERIAN THEOREMS;

FINANCE;

EID: 30344447274     PISSN: 03044149     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.spa.2005.09.002     Document Type: Article

References (33)

1. S. Asmussen Subexponential asymptotics for stochastic processes: extremal behavior, stationary distributions and first passage probabilities Ann. Appl. Probab. 8 2 1998 354 374 ISSN 1050-5164
2. J. Bertoin Lévy processes Cambridge Tracts in Mathematics vol. 121 1996 Cambridge University Press Cambridge ISBN 0-521-56243-0
3. J. Bertoin, P. Biane, M. Yor, Poissonian exponential functionals, q-series, q-integrals, and the moment problem for log-normal distributions, in: Seminar on Stochastic Analysis, Random Fields and Applications IV, Progress Probability, Birkhäuser, Basel, 2004, pp. 45-56.
4. J. Bertoin, and M. Yor On the entire moments of self-similar Markov processes and exponential functionals of Lévy processes Ann. Fac. Sci. Toulouse Math. (6) 11 1 2002 33 45 ISSN 0240-2963
5. N.H. Bingham, C.M. Goldie, and J.L. Teugels Regular variation Encyclopedia of Mathematics and its Applications vol. 27 1987 Cambridge University Press Cambridge ISBN 0-521-30787-2
6. J. Blanchet, P. Glynn, Approximations for the distribution of infinite horizon discounted rewards, Technical Report, Department of Management Science and Engineering, Stanford University, 2004.
7. M. Braverman, T. Mikosch, and G. Samorodnitsky Tail probabilities of subadditive functionals of Lévy processes Ann. Appl. Probab. 12 1 2002 69 100 ISSN 1050-5164.
8. L. Breiman On some limit theorems similar to the arc-sin law Teor. Verojatnost. i Primenen. 10 1965 351 360
9. P. Carmona, F. Petit, M. Yor, On the distribution and asymptotic results for exponential functionals of Lévy processes, in: Exponential functionals and principal values related to Brownian motion, Bibl. Rev. Mat. Iberoamericana, Rev. Mat. Iberoamericana, Madrid, 1997, pp. 73-130.
10. D. Denisov, S. Foss, and D. Korshunov Tail asymptotics for the supremum of a random walk when the mean is not finite Queueing Syst. 46 1-2 2004 15 33 ISSN 0257-0130
11. D. Dufresne, The distribution of a perpetuity, with applications to risk theory and pension funding, Scand. Actuar. J. (1-2) (1990) 39-79 ISSN 0346-1238.
12. V. Dumas, F. Guillemin, and P. Robert A Markovian analysis of additive-increase multiplicative-decrease algorithms Adv. Appl. Probab. 34 1 2002 85 111 ISSN 0001-8678
13. P. Embrechts, C.M. Goldie, and N. Veraverbeke Subexponentiality and infinite divisibility Z. Wahrsch. Verw. Gebiete 49 3 1979 335 347 ISSN 0044-3719
14. P. Embrechts, C. Klüppelberg, and T. Mikosch Modelling extremal events for insurance and finance Applications of Mathematics (New York) vol. 33 1997 Springer Berlin ISBN 3-540-60931-8
15. K.B. Erickson, R.A. Maller, Generalised Ornstein-Uhlenbeck processes and the convergence of Lévy integrals, in: Séminaire de Probabilités XXXVIII, Lecture Notes in Mathematics, vol. 1857, Springer, Berlin, 2005, pp. 70-94.
16. S. Foss, T. Konstantapoulos, S. Zachary, The principle of a single big jump: discrete and continuous time modulated random walks with heavy-tailed increments, 2005, preprint available as arXiv:math.PR/0509605 at http://arxiv.org/PS_cache/math/pdf/0509/0509605.pdf.
17. S. Foss, Z. Palmowski, and S. Zachary The probability of exceeding a high boundary on a random time interval for a heavy-tailed random walk Ann. Appl. Probab. 15 3 2005 1936 1957 ISSN 1050-5164
18. C.M. Goldie Implicit renewal theory and tails of solutions of random equations Ann. Appl. Probab. 1 1 1991 126 166 ISSN 1050-5164
19. C.M. Goldie, and R. Grübel Perpetuities with thin tails Adv. Appl. Probab. 28 2 1996 463 480 ISSN 0001-8678
20. C.M. Goldie, and R.A. Maller Stability of perpetuities Ann. Probab. 28 3 2000 1195 1218 ISSN 0091-1798
21. F. Guillemin, P. Robert, and B. Zwart AIMD algorithms and exponential functionals Ann. Appl. Probab. 14 1 2004 90 117 ISSN 1050-5164
22. H. Kesten Random difference equations and renewal theory for products of random matrices Acta Math. 131 1973 207 248
23. 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 ISSN 1050-5164
24. C. Klüppelberg, A. Lindner, R. Maller, Continuous time volatility modelling: COGARCH versus Ornstein-Uhlenbeck models, in: Y. Kabanov, R. Lipster, J. Stoyanov (Eds.), From stochastic calculus to mathematical finance. The Shiryaev Festschrift, Springer, to appear. Available at http://www-m4.ma.tum.de/m4/Papers/Klueppelberg/cog-spr.pdf.
25. D.G. Konstantinides, T. Mikosch, Large deviations and ruin probabilities for solutions to stochastic recurrence equations with heavy-tailed innovations, Ann. Probab. 33 (5) (2005) 1992-2035 ISSN 0091-1798.
26. N. Litvak, and W.R. van Zwet On the minimal travel time needed to collect n items on circle Ann. Appl. Probab. 14 2 2004 881 902 ISSN 1050-5164
27. V. Rivero A law of iterated logarithm for increasing self-similar Markov processes Stochastic Stochastic Rep. 75 6 2003 443 472 ISSN 1045-1129
28. V. Rivero Recurrent extensions of self-similar Markov processes and Cramér's condition Bernoulli 11 3 2005 471 509 ISSN 1350-7265
29. H. Rootzén Extreme value theory for moving average processes Ann. Probab. 14 2 1986 612 652 ISSN 0091-1798
30. N. Veraverbeke Asymptotic behaviour of Wiener-Hopf factors of a random walk Stochastic Processes Appl. 5 1 1977 27 37
31. E. Willekens On the supremum of an infinitely divisible process Stochastic Process. Appl. 26 1 1987 173 175 ISSN 0304-4149
32. M. Yor, Exponential functionals of Brownian motion and related processes, Springer Finance. Springer, Berlin, 2001. ISBN 3-540-65943-9. (With an introductory chapter by Hélyette Geman, Chapters 1, 3, 4, 8 translated from the French by Stephen S. Wilson).
33. S. Zachary A note on Veraverbeke's theorem Queueing Syst. 46 1-2 2004 9 14 ISSN 0257-0130