Acta Applicandae Mathematicae

, Volume 123, Issue 1, pp 113–139 | Cite as

Fluctuations of Stable Processes and Exponential Functionals of Hypergeometric Lévy Processes

Article

Abstract

We study the distribution and various properties of exponential functionals of hypergeometric Lévy processes. We derive an explicit formula for the Mellin transform of the exponential functional and give both convergent and asymptotic series expansions of its probability density function. As applications we present a new proof of some of the results on the density of the supremum of a stable process, which were recently obtained in Hubalek and Kuznetsov (Electron. Commun. Probab. 16:84–95, 2011) and Kuznetsov (Ann. Probab. 39(3):1027–1060, 2011). We also derive several new results related to (i) the entrance law of a stable process conditioned to stay positive, (ii) the entrance law of the excursion measure of a stable process reflected at its past infimum, (iii) the distribution of the lifetime of a stable process conditioned to hit zero continuously and (iv) the entrance law and the last passage time of the radial part of a multidimensional symmetric stable process.

Keywords

Hypergeometric Lévy processes Lamperti-stable processes Exponential functional Double gamma function Lamperti transformation Extrema of stable processes 

Mathematics Subject Classification

60G51 60G52 

References

  1. 1.
    Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables. Dover, New York (1970) Google Scholar
  2. 2.
    Barnes, E.W.: The genesis of the double gamma function. Proc. Lond. Math. Soc. 31, 358–381 (1899) MATHCrossRefGoogle Scholar
  3. 3.
    Barnes, E.W.: The theory of the double gamma function. Philos. Trans. R. Soc. Lond. A 196, 265–387 (1901) MATHCrossRefGoogle Scholar
  4. 4.
    Bernyk, V., Dalang, R.C., Peskir, G.: The law of the supremum of a stable Lévy process with no negative jumps. Ann. Probab. 36(5), 1777–1789 (2008) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Bertoin, J.: Lévy Processes. Cambridge Tracts in Mathematics, vol. 121. Cambridge University Press, Cambridge (1996) MATHGoogle Scholar
  6. 6.
    Bertoin, J., Yor, M.: The entrance laws of self-similar Markov processes and exponential functionals of Lévy processes. Potential Anal. 17(4), 389–400 (2002) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Bertoin, J., Yor, M.: Exponential functionals of Lévy processes. Probab. Surv. 2, 191–212 (2005) (electronic) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Billingham, J., King, A.C.: Uniform asymptotic expansions for the Barnes double gamma function. Proc. R. Soc. Lond. A 453, 1817–1829 (1997) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Caballero, M.E., Chaumont, L.: Conditioned stable Lévy processes and the Lamperti representation. J. Appl. Probab. 43(4), 967–983 (2006) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Caballero, M.E., Chaumont, L.: Weak convergence of positive self-similar Markov processes and overshoots of Lévy processes. Ann. Probab. 34(3), 1012–1034 (2006) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Caballero, M., Rivero, V.: On the asymptotic behaviour of increasing self-similar Markov processes. Electron. J. Probab. 14, 865–894 (2009) (electronic) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Caballero, M.E., Pardo, J.C., Pérez, J.L.: On Lamperti stable processes. Probab. Math. Stat. 30, 1–28 (2010) MATHGoogle Scholar
  13. 13.
    Caballero, M.E., Pardo, J.C., Pérez, J.L.: Explicit identities for Lévy processes associated to symmetric stable processes. Bernoulli 17(1), 34–59 (2011) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Cai, N., Kou, S.G.: Pricing Asian options under a hyper-exponential jump diffusion model. Oper. Res. 60(1), 64–77 (2012). doi:10.1287/opre.1110.1006 MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Carmona, P., Petit, F., Yor, M.: On the distribution and asymptotic results for exponential functionals of Levy processes. In: Yor, M. (ed.) Exponential Functionals and Principal Values Related to Brownian Motion, Bibl. Rev. Mat. Iberoamericana, pp. 73–121 (1997) Google Scholar
  16. 16.
    Chaumont, L.: Conditionings and path decompositions for Lévy processes. Stoch. Process. Appl. 64(1), 39–54 (1996) MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Chaumont, L.: On the law of the supremum of Lévy processes. Ann. Probab. (2012, to appear) Google Scholar
  18. 18.
    Chaumont, L., Doney, R.A.: On Lévy processes conditioned to stay positive. Electron. J. Probab. 28(10), 948–961 (2005) (electronic) MathSciNetGoogle Scholar
  19. 19.
    Chaumont, L., Pardo, J.C.: The lower envelope of positive self-similar Markov processes. Electron. J. Probab. 11(49), 1321–1341 (2006) (electronic) MathSciNetMATHGoogle Scholar
  20. 20.
    Chaumont, L., Kyprianou, A.E., Pardo, J.C.: Some explicit identities associated with positive self-similar Markov processes. Stoch. Process. Appl. 119(3), 980–1000 (2009) MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Conway, J.B.: Functions of One Complex Variable, 2nd edn. Springer, Berlin (1978) CrossRefGoogle Scholar
  22. 22.
    Doney, R.A.: On Wiener-Hopf factorisation and the distribution of extrema for certain stable processes. Ann. Probab. 15(4), 1352–1362 (1987) MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Getoor, R.: The Brownian escape process. Ann. Probab. 7(5), 864–867 (1979) MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Haas, B.: Loss of mass in deterministic and random fragmentations. Stoch. Process. Appl. 106(2), 245–277 (2003) MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Hubalek, F., Kuznetsov, A.: A convergent series representation for the density of the supremum of a stable process. Electron. Commun. Probab. 16, 84–95 (2011) MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Jeffrey, A. (ed.): Table of Integrals, Series and Products, 7th edn. Academic Press, San Diego (2007) MATHGoogle Scholar
  27. 27.
    Kuznetsov, A.: On extrema of stable processes. Ann. Probab. 39(3), 1027–1060 (2011) MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Kuznetsov, A.: On the density of the supremum of a stable process (2012). arXiv:1112.4208
  29. 29.
    Kuznetsov, A., Kyprianou, A.E., Pardo, J.C., Van Schaik, K.: A Wiener-Hopf Monte-Carlo simulation technique for Lévy processes. Ann. Appl. Probab. 21(6), 2171–2190 (2011) MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Kuznetsov, A., Kyprianou, A.E., Pardo, J.C.: Meromorphic Lévy processes and their fluctuation identities. Ann. Appl. Probab. (2012, to appear) Google Scholar
  31. 31.
    Kyprianou, A.: Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin (2006) MATHGoogle Scholar
  32. 32.
    Kyprianou, A.E., Pardo, J.C.: Continuous-state branching processes and self-similarity. J. Appl. Probab. 45(4), 1140–1160 (2008) MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Kyprianou, A.E., Pardo, J.C., Rivero, V.M.: Exact and asymptotic n-tuple laws at first and last passage. Ann. Appl. Probab. 20(2), 522–564 (2010) MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Lamperti, J.: Semi-stable Markov processes. I. Z. Wahrscheinlichkeitstheor. Verw. Geb. 22, 205–225 (1972) MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Lawrie, J.B., King, A.C.: Exact solutions to a class of functional difference equations with application to a moving contact line flow. Eur. J. Appl. Math. 5, 141–147 (1994) MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    Maulik, K., Zwart, B.: Tail asymptotics for exponential functionals of Lévy processes. Stoch. Process. Appl. 116, 156–177 (2006) MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    Pardo, J.C.: On the future infimum of positive self-similar Markov processes. Stochastics 78(3), 123–155 (2006) MathSciNetMATHGoogle Scholar
  38. 38.
    Patie, P.: Exponential functionals of a new family of Lévy processes and self-similar continuous state branching processes with immigration. Bull. Sci. Math. 133(4), 355–382 (2009) MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    Patie, P.: A few remarks on the supremum of stable processes. Stat. Probab. Lett. 79, 1125–1128 (2009) MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    Patie, P.: Law of the absorption time of positive self-similar Markov processes. Ann. Probab. 40(2), 765–787 (2012). doi:10.1214/10-AOP638 MathSciNetMATHCrossRefGoogle Scholar
  41. 41.
    Rivero, V.: A law of iterated logarithm for increasing self-similar Markov processes. Stoch. Stoch. Rep. 75(6), 443–472 (2003) MathSciNetMATHCrossRefGoogle Scholar
  42. 42.
    Rivero, V.: Recurrent extensions of self-similar Markov processes and Cramér’s condition. Bernoulli 11(3), 471–509 (2005) MathSciNetMATHCrossRefGoogle Scholar
  43. 43.
    Rivero, V.: Recurrent extensions of self-similar Markov processes and Cramér’s condition. II. Bernoulli 13(4), 1053–1070 (2007) MathSciNetMATHCrossRefGoogle Scholar
  44. 44.
    Sato, K.: Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics, vol. 68. Cambridge University Press, Cambridge (1999) MATHGoogle Scholar
  45. 45.
    Vigon, V.: Simplifiez vos Lévy en titillant la factorisation de Wiener-Hopf. Thèse de doctorat de l’INSA de Rouen (2002) Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsYork UniversityTorontoCanada
  2. 2.Centro de Investigación en MatemáticasGuanajuatoMéxico

Personalised recommendations