Skip to main content
SpringerLink
Log in

Fluctuations of Stable Processes and Exponential Functionals of Hypergeometric Lévy Processes

  • Published:
Acta Applicandae Mathematicae Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables. Dover, New York (1970)

    Google Scholar 

  2. Barnes, E.W.: The genesis of the double gamma function. Proc. Lond. Math. Soc. 31, 358–381 (1899)

    Article  MATH  Google Scholar 

  3. Barnes, E.W.: The theory of the double gamma function. Philos. Trans. R. Soc. Lond. A 196, 265–387 (1901)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bertoin, J.: Lévy Processes. Cambridge Tracts in Mathematics, vol. 121. Cambridge University Press, Cambridge (1996)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bertoin, J., Yor, M.: Exponential functionals of Lévy processes. Probab. Surv. 2, 191–212 (2005) (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  8. Billingham, J., King, A.C.: Uniform asymptotic expansions for the Barnes double gamma function. Proc. R. Soc. Lond. A 453, 1817–1829 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  9. Caballero, M.E., Chaumont, L.: Conditioned stable Lévy processes and the Lamperti representation. J. Appl. Probab. 43(4), 967–983 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  11. Caballero, M., Rivero, V.: On the asymptotic behaviour of increasing self-similar Markov processes. Electron. J. Probab. 14, 865–894 (2009) (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  12. Caballero, M.E., Pardo, J.C., Pérez, J.L.: On Lamperti stable processes. Probab. Math. Stat. 30, 1–28 (2010)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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. Chaumont, L.: Conditionings and path decompositions for Lévy processes. Stoch. Process. Appl. 64(1), 39–54 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  17. Chaumont, L.: On the law of the supremum of Lévy processes. Ann. Probab. (2012, to appear)

  18. Chaumont, L., Doney, R.A.: On Lévy processes conditioned to stay positive. Electron. J. Probab. 28(10), 948–961 (2005) (electronic)

    MathSciNet  Google Scholar 

  19. Chaumont, L., Pardo, J.C.: The lower envelope of positive self-similar Markov processes. Electron. J. Probab. 11(49), 1321–1341 (2006) (electronic)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  21. Conway, J.B.: Functions of One Complex Variable, 2nd edn. Springer, Berlin (1978)

    Book  Google Scholar 

  22. Doney, R.A.: On Wiener-Hopf factorisation and the distribution of extrema for certain stable processes. Ann. Probab. 15(4), 1352–1362 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  23. Getoor, R.: The Brownian escape process. Ann. Probab. 7(5), 864–867 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  24. Haas, B.: Loss of mass in deterministic and random fragmentations. Stoch. Process. Appl. 106(2), 245–277 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  26. Jeffrey, A. (ed.): Table of Integrals, Series and Products, 7th edn. Academic Press, San Diego (2007)

    MATH  Google Scholar 

  27. Kuznetsov, A.: On extrema of stable processes. Ann. Probab. 39(3), 1027–1060 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  28. Kuznetsov, A.: On the density of the supremum of a stable process (2012). arXiv:1112.4208

  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)

    Article  MathSciNet  MATH  Google Scholar 

  30. Kuznetsov, A., Kyprianou, A.E., Pardo, J.C.: Meromorphic Lévy processes and their fluctuation identities. Ann. Appl. Probab. (2012, to appear)

  31. Kyprianou, A.: Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin (2006)

    MATH  Google Scholar 

  32. Kyprianou, A.E., Pardo, J.C.: Continuous-state branching processes and self-similarity. J. Appl. Probab. 45(4), 1140–1160 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  34. Lamperti, J.: Semi-stable Markov processes. I. Z. Wahrscheinlichkeitstheor. Verw. Geb. 22, 205–225 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  36. Maulik, K., Zwart, B.: Tail asymptotics for exponential functionals of Lévy processes. Stoch. Process. Appl. 116, 156–177 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  37. Pardo, J.C.: On the future infimum of positive self-similar Markov processes. Stochastics 78(3), 123–155 (2006)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  39. Patie, P.: A few remarks on the supremum of stable processes. Stat. Probab. Lett. 79, 1125–1128 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  41. Rivero, V.: A law of iterated logarithm for increasing self-similar Markov processes. Stoch. Stoch. Rep. 75(6), 443–472 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  42. Rivero, V.: Recurrent extensions of self-similar Markov processes and Cramér’s condition. Bernoulli 11(3), 471–509 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  43. Rivero, V.: Recurrent extensions of self-similar Markov processes and Cramér’s condition. II. Bernoulli 13(4), 1053–1070 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  44. Sato, K.: Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics, vol. 68. Cambridge University Press, Cambridge (1999)

    MATH  Google Scholar 

  45. Vigon, V.: Simplifiez vos Lévy en titillant la factorisation de Wiener-Hopf. Thèse de doctorat de l’INSA de Rouen (2002)

Download references

Acknowledgements

A. Kuznetsov’s research is supported by the Natural Sciences and Engineering Research Council of Canada.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, Ontario, M3J 1P3, Canada

    A. Kuznetsov

  2. Centro de Investigación en Matemáticas, A.C. Calle Jalisco s/n, 36240, Guanajuato, México

    J. C. Pardo

Authors
  1. A. Kuznetsov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. C. Pardo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. C. Pardo.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kuznetsov, A., Pardo, J.C. Fluctuations of Stable Processes and Exponential Functionals of Hypergeometric Lévy Processes. Acta Appl Math 123, 113–139 (2013). https://doi.org/10.1007/s10440-012-9718-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10440-012-9718-y

Keywords

Mathematics Subject Classification

Search

Navigation