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 Electronic Journal of Probability > Vol. 13 (2008) > Paper 57 open journal systems 


Special, conjugate and complete scale functions for spectrally negative Lévy processes

Andreas E Kyprianou, University of Bath
Victor Rivero, Centro de Investigación en Matemáticas, Mexico


Abstract
Following from recent developments in Hubalek and Kyprianou [28], the objective of this paper is to provide further methods for constructing new families of scale functions for spectrally negative Lévy processes which are completely explicit. This is the result of an observation in the aforementioned paper which permits feeding the theory of Bernstein functions directly into the Wiener-Hopf factorization for spectrally negative Lévy processes. Many new, concrete examples of scale functions are offered although the methodology in principle delivers still more explicit examples than those listed.


Full text: PDF

Pages: 1672-1701

Published on: September 21, 2008
Bibliography
  1. F. Avram ; T. Chan ; M. Usabel . On the valuation of constant barrier options under spectrally one-sided exponential Lévy models and Carr's approximation for American puts. Stochastic Process. Appl. 100 (2002), 75--107. MR1919609 (2003d:91041)
  2. F. Avram ; A. E. Kyprianou ; M. R. Pistorius . Exit problems for spectrally negative Lévy processes and applications to (Canadized) Russian options. Ann. Appl. Probab. 14 (2004), no. 1, 215--238. MR2023021 (2005c:60053)
  3. F. Avram ; Z. Palmowski ; M. R. Pistorius . On the optimal dividend problem for a spectrally negative Lévy process. Ann. Appl. Probab. 17 (2007), no. 1, 156--180. MR2292583 (2008e:93121)
  4. R. Bekker ; O. Boxma ; O. Kella . Queues with delays in two state strategies and Lévy input. EURANDOM, report 2007-08 (2007). Math. Review number not available.
  5. C. Berg ; G. Forst . Potential theory on locally compact abelian groups. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 87. Springer-Verlag, New York-Heidelberg, 1975. vii+197 pp. MR0481057 (58 #1204)
  6. J. Bertoin . Lévy processes. Cambridge Tracts in Mathematics, 121. Cambridge University Press, Cambridge, 1996. x+265 pp. ISBN: 0-521-56243-0 MR1406564 (98e:60117)
  7. J. Bertoin . Exponential decay and ergodicity of completely asymmetric Lévy processes in a finite interval. Ann. Appl. Probab. 7 (1997), no. 1, 156--169. MR1428754 (98d:60147)
  8. J. Bertoin . Regenerative embedding of Markov sets. Probab. Theory Related Fields 108 (1997), no. 4, 559--571. MR1465642 (98h:60013)
  9. J. Bertoin ; B. Roynette ; M. Yor . Some connections between (sub)critical branching mechanisms and Bernstein functions. Preprint (2004). Math. Review number not available.
  10. N. H. Bingham . Continuous branching processes and spectral positivity. Stochastic Processes Appl. 4 (1976), no. 3, 217--242. MR0410961 (53 #14701)
  11. L. Bondesson . Generalized gamma convolutions and related classes of distributions and densities. Lecture Notes in Statistics, 76. Springer-Verlag, New York, 1992. viii+173 pp. ISBN: 0-387-97866-6 MR1224674 (94g:60031)
  12. L. Chaumont Sur certains processus de Lévy conditionnés à rester positifs. (French) [On certain Levy processes that are conditioned to stay positive] Stochastics Stochastics Rep. 47 (1994), no. 1-2, 1--20. MR1787140 (2001e:60093)
  13. L. Chaumont Conditionings and path decompositions for Lévy processes. Stochastic Process. Appl. 64 (1996), no. 1, 39--54. MR1419491 (98b:60131)
  14. L. Chaumont ; A. E. Kyprianou ; J. C. Pardo. Some explicit identities associated with positive self-similar Markov processes. Preprint (2007). Math. Review number not available.
  15. S. N. Chiu ; C. Yin . Passage times for a spectrally negative Lévy process with applications to risk theory. Bernoulli 11 (2005), no. 3, 511--522. MR2146892 (2006g:60074)
  16. C. Donati-Martin ; M. Yor . Further examples of explicit Krein representations of certain subordinators. Publ. Res. Inst. Math. Sci. 43 (2007), no. 2, 315--328. MR2341013 (2008f:60050)
  17. R. A. Doney . Hitting probabilities for spectrally positive Lévy processes. J. London Math. Soc. (2) 44 (1991), no. 3, 566--576. MR1149016 (93b:60166)
  18. R. A. Doney . Some excursion calculations for spectrally one-sided Lévy processes. Séminaire de Probabilités XXXVIII, 5--15, Lecture Notes in Math., 1857, Springer, Berlin, 2005. MR2126963 (2006b:60098)
  19. 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. Edited and with a foreword by Jean Picard. Lecture Notes in Mathematics, 1897. Springer, Berlin, 2007. x+147 pp. ISBN: 978-3-540-48510-0; 3-540-48510-4 MR2320889
  20. R. A. Doney ; A. E. Kyprianou . Overshoots and undershoots of Lévy processes. Ann. Appl. Probab. 16 (2006), no. 1, 91--106. MR2209337 (2007b:60117)
  21. P. Dube ; F. Guillemin ; R. R. Mazumdar . Scale functions of Lévy processes and busy periods of finite-capacity $M/GI/1$ queues. J. Appl. Probab. 41 (2004), no. 4, 1145--1156. MR2122808 (2005m:60208)
  22. D. J. Emery . Exit problem for a spectrally positive process. Advances in Appl. Probability 5 (1973), 498--520. MR0341623 (49 #6370)
  23. H. Furrer . Risk processes perturbed by $alpha$-stable Lévy motion. Scand. Actuar. J. (1998) no. 1, 59--74. MR1626676 (99e:60198)
  24. J. Hawkes . On the potential theory of subordinators. Z. Warscheinlichkeitstheorie und Verw. Gebiete 33 (1975/76), no. 2, 113--132. MR0388553 (52 #9389)
  25. J. Hawkes . Intersections of Markov random sets. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 37 (1976/77), no. 3, 243--251. MR0483035 (58 #3064)
  26. B. Hilberink ; L. C. G. Rogers . Optimal capital structure and endogenous default. Finance Stoch. 6 (2002), no. 2, 237--263. MR1897961
  27. F. Hirsch . Familles d'opérateurs potentiels. (French) Collection of articles dedicated to Marcel Brelot on the occasion of his 70th birthday. Ann. Inst. Fourier (Grenoble) 25 (1975), no. 3-4, xxii, 263--288. MR0402088 (53 #5911)
  28. F. Hubalek ; A. E. Kyprianou . Old and new examples of scale functions for spectrally negative Lévy processes. Preprint (2007). Math. Review number not available.
  29. T. Huillet . Energy cascades as branching processes with emphasis on Neveu's approach to Derrida's random energy model. Adv. in Appl. Probab. 35 (2003), no. 2, 477--503. MR1970484 (2005k:60236)
  30. N. Jacob Pseudo differential operators and Markov processes. Vol. I. Fourier analysis and semigroups. Imperial College Press, London, 2001. xxii+493 pp. ISBN: 1-86094-293-8 MR1873235 (2003a:47104)
  31. L. F. James ; B. Roynette ; M. Yor . Generalized gamma convolutions, Dirichlet means, Thorin measures, with explicit examples. Preprint (2007). Math. Review number not available.
  32. J. F. C. Kingman . Markov transition probabilities. I. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 7 (1967) 248--270. MR0217886 (36 #975)
  33. C. Klüppelberg ; A. E. Kyprianou . On extreme ruinous behaviour of Lévy insurance risk processes. J. Appl. Probab. 43 (2006), no. 2, 594--598. MR2248586 (2007k:91169)
  34. C. Klüppelberg ; A. E. Kyprianou ; R. A. Maller . Ruin probabilities and overshoots for general Lévy insurance risk processes. Ann. Appl. Probab. 14 (2004), no. 4, 1766--1801. MR2099651 (2005j:60097)
  35. V. S. Koroljuk . Boundary problems for a compound Poisson process. (Russian) Teor. Verojatnost. i Primenen. 19 (1974), 3--14. MR0336824 (49 #1597)
  36. V. S.Koroljuk . Ruin problems for a compound Poisson process. (Russian) Teor. Verojatnost. i Primenen. 20 (1975), no. 2, 382--384. MR0402940 (53 #6754)
  37. N. Krell . Multifractal spectra and precise rates of decay in homogeneous fragmentations. Stochastic Process. Appl. 118 (2008), no. 6, 897--916. MR2418249
  38. A. E. Kyprianou . Introductory lectures on fluctuations of Lévy processes with applications. Universitext. Springer-Verlag, Berlin, 2006. xiv+373 pp. ISBN: 978-3-540-31342-7; 3-540-31342-7 MR2250061 (2008a:60003)
  39. A. E. Kyprianou ; Z. Palmowski . Quasi-stationary distributions for Lévy processes. Bernoulli 12 (2006), no. 4, 571--581. MR2248228 (2007k:60136)
  40. A. E. Kyprianou ; Z. Palmowski . Distributional study of de Finetti's dividend problem for a general Lévy insurance risk process. J. Appl. Probab. 44 (2007), no. 2, 428--443. MR2340209
  41. A. E. Kyprianou ; B. A. Surya . Principles of smooth and continuous fit in the determination of endogenous bankruptcy levels. Finance Stoch. 11 (2007), no. 1, 131--152. MR2284015 (2008b:91069)
  42. A. Lambert . Completely asymmetric Lévy processes confined in a finite interval. Ann. Inst. H. Poincaré Probab. Statist. 36 (2000), no. 2, 251--274. MR1751660 (2001f:60049)
  43. A. Lambert . Quasi-stationary distributions and the continuous-state branching process conditioned to be never extinct. Electron. J. Probab. 12 (2007), no. 14, 420--446 (electronic). MR2299923 (2008b:60183)
  44. R. Loeffen . On optimality of the barrier strategy in De Finetti's dividend problem for spectrally negative Lévy processes. To appear in Ann. Appl. Probab. (2008). Math. Review number not available.
  45. P. Patie .Law of the exponential functional of a new family of one-sided Lévy processes via self-similar continuous state branching processes with immigration and the wright hypergeometric functions. Preprint (2007). Math. Review number not available.
  46. M. R. Pistorius . On doubly reflected completely asymmetric Lévy processes. Stochastic Process. Appl. 107 (2003), no. 1, 131--143. MR1995924 (2004k:60137)
  47. M. R. Pistorius . On exit and ergodicity of the spectrally one-sided Lévy process reflected at its infimum. J. Theoret. Probab. 17 (2004), no. 1, 183--220. MR2054585 (2005e:60104)
  48. M. R. Pistorius . A potential-theoretical review of some exit problems of spectrally negative Lévy processes. Séminaire de Probabilités XXXVIII, 30--41, Lecture Notes in Math., 1857, Springer, Berlin, 2005. MR2126965 (2006k:60086)
  49. M. R. Pistorius . An excursion-theoretical approach to some boundary crossing problems and the Skorokhod embedding for reflected Lévy processes. Séminaire de Probabilités XL, 287--307, Lecture Notes in Math., 1899, Springer, Berlin, 2007. MR2409012
  50. H. Pollard . The completely monotonic character of the Mittag-Leffler function $Esb a(-x)$. Bull. Amer. Math. Soc. 54, (1948). 1115--1116. MR0027375 (10,295e)
  51. J. F. Renaud ; X. Zhou . Distribution of the present value of dividend payments in a Lévy risk model. J. Appl. Probab. 44 (2007), no. 2, 420--427. MR2340208
  52. L. C. G. Rogers . The two-sided exit problem for spectrally positive Lévy processes. Adv. in Appl. Probab. 22 (1990), no. 2, 486--487. MR1053243 (93f:60110)
  53. J. Rosinski . Tempering stable processes. Stochastic Process. Appl. 117 (2007), no. 6, 677--707. MR2327834 (2008g:60146)
  54. K. Sato . Lévy processes and infinitely divisible distributions. Translated from the 1990 Japanese original. Revised by the author. Cambridge Studies in Advanced Mathematics, 68. Cambridge University Press, Cambridge, 1999. xii+486 pp. ISBN: 0-521-55302-4 MR1739520 (2003b:60064)
  55. W. R. Schneider Completely monotone generalized Mittag-Leffler functions. Exposition. Math. 14 (1996), no. 1, 3--16. MR1382012 (97a:33041)
  56. R. Song ; Z. Vondracek . Potential theory of special subordinators and subordinate killed stable processes. J. Theoret. Probab. 19 (2006), no. 4, 817--847. MR2279605 (2008g:60237)
  57. R. Song ; Z. Vondracek . Potential theory of subordinate Brownian motion. Lecture notes (2007). Math. Review number not available.
  58. R. Song ; Z. Vondracek . Some remarks on special subordinators. To appear in Rocky Mountain Journal of Mathematics (2008). Math. Review number not available.
  59. V. N. Suprun . The ruin problem and the resolvent of a killed independent increment process. (Russian) Ukrain. Mat. v Z. 28 (1976), no. 1, 53--61, 142. MR0428476 (55 #1497)
  60. L. Takács . Combinatorial methods in the theory of stochastic processes. Reprint of the 1967 original. Robert E. Krieger Publishing Co., Huntington, N. Y., 1977. xi+262 pp. MR0431313 (55 #4314)
  61. O. Thorin . On the infinite divisibility of the lognormal distribution. Scand. Actuar. J. (1977) no. 3, 121--148. MR0552135 (80m:60022)
  62. V. Vigon . Votre Lévy rampe-t-il? (French) [Does your Levy process creep?] J. London Math. Soc. (2) 65 (2002), no. 1, 243--256. MR1875147 (2002i:60101)
  63. V. M. Zolotarev . The moment of first passage of a level and the behaviour at infinity of a class of processes with independent increments. (Russian) Teor. Verojatnost. i Primenen. 9 (1964) 724--733. MR0171315 (30 #1546)
















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Electronic Journal of Probability. ISSN: 1083-6489