Skip to main content
SpringerLink
Log in

On Exponential Functionals of Lévy Processes

  • Published:
Journal of Theoretical Probability Aims and scope Submit manuscript

Abstract

Exponential functionals of Lévy processes appear as stationary distributions of generalized Ornstein–Uhlenbeck (GOU) processes. In this paper we obtain the infinitesimal generator of the GOU process and show that it is a Feller process. Further, we use these results to investigate properties of the mapping \(\Phi \), which maps two independent Lévy processes to their corresponding exponential functional, where one of the processes is assumed to be fixed. We show that in many cases this mapping is injective, and give the inverse mapping in terms of (Lévy) characteristics. Also, continuity of \(\Phi \) is treated, and some results on its range are obtained.

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. Alsmeyer, G., Iksanov, A., Rösler, U.: On distributional properties of perpetuities. J. Theoret. Probab. 22, 666–682 (2009)

    Article  MathSciNet  Google Scholar 

  2. Applebaum, D.: Lévy Processes and Stochastic Calculus. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  3. Behme, A.: Distributional properties of solutions of \(dV_t = V_{t-} dU_t + dL_t\) with Lévy noise. Adv. Appl. Prob. 43, 688–711 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Behme, A., Lindner, A., Maller, R.: Stationary solutions of the stochastic differential equation \(dV_t = V_{t-} dU_t + dL_t\) with Lévy noise. Stoch. Process. Appl. 121, 91–108 (2011)

    Article  MathSciNet  Google Scholar 

  5. Behme, A., Maejima, M., Matsui, M., Sakuma, N.: Distributions of exponential integrals of independent increment processes related to generalized gamma convolutions. Bernoulli 18, 1172–1187 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bertoin, J., Lindner, A., Maller, R.: On continuity properties of the law of integrals of Lévy processes. In: Donati-Martin, C., Émery, M., Rouault, A., Stricker, C. (eds.) Séminaire de Probabilités XLI, Lecture Notes in Mathematics 1934, 137–159. Springer, Berlin (2008)

    Google Scholar 

  7. Billingsley, P.: Convergence of Probability Measures, 2nd edn. Wiley Series in Probability and Statistics, New York (1999)

    Book  MATH  Google Scholar 

  8. Brandt, A.: The stochastic equation \(Y_{n+1}=A_nY_n+B_n\) with stationary coefficients. Adv. Appl. Prob. 18, 211–220 (1986)

    Article  MATH  Google Scholar 

  9. Carmona, P.: Some complements to “On the distribution and asymptotic results for exponential functionals of Lévy processes” (1996). Available online at http://www.math.sciences.univ-nantes.fr/~carmona/com.ps.gz

  10. Carmona, P., Petit, F., Yor, M.: On the distribution and asymptotic results for exponential functionals of Lévy processes. In Yor, M. (ed.): Exponential Functionals and Principal Values Related to Brownian Motion, Bibl. Rev. Mat. Iberoamericana, Rev. Mat. Iberoamericana, Madrid, pp. 73–130 (1997)

  11. Carmona, P., Petit, F., Yor, M.: Exponential functionals of Lévy processes. In: Barndorff-Nielsen, O.E., Mikosch, T., Resnick, S. (eds.) Lévy Processes, pp. 41–55. Birkhäuser Boston, Boston (2001)

    Chapter  Google Scholar 

  12. Erickson, K.B., Maller, R.A.: Generalised Ornstein-Uhlenbeck processes and the convergence of Lévy integrals. In: Emery, M., Ledoux, M., Yor, M. (eds.) Séminaire de Probabilités XXXVIII, Lecture Notes in Mathematics 1857, 70–94. Springer, Berlin (2005)

    Google Scholar 

  13. Ethier, S., Kurtz, T.: Markov Processes: Characterization and Convergence. Wiley Series in Probability and Mathematical Statistics, New York (1986)

    Book  MATH  Google Scholar 

  14. Il’inskii, A.I.: On the zeros and argument of characteristic functions. Theory Probab. Appl. 29, 410–415 (1975)

    Google Scholar 

  15. Jacod, J., Shiryaev, A.N.: Limit Theorems for Stochastic Processes, 2nd edn. Springer, New-York (2003)

    Book  MATH  Google Scholar 

  16. Jurek, Z.J., Mason, J.D.: Operator-Limit Distributions in Probability Theory. Wiley, New York (1993)

    MATH  Google Scholar 

  17. Kallenberg, O.: Foundations of Modern Probability, 2nd edn. Springer, Berlin (2001)

    Google Scholar 

  18. Karandikar, R.L.: Multiplicative decomposition of nonsingular matrix valued semimartingales. In: Azéma, J., Yor, M., Meyer, P. (eds.) Séminaire de Probabilités XXV, Lecture Notes in Math, vol. 1485, pp. 262–269, Springer, Berlin (1991)

  19. Kato, T.: Perturbation Theory for Linear Operators, 2nd edn. Springer, Berlin (1976)

    Book  MATH  Google Scholar 

  20. Kawata, T.: Fourier Analysis in Probability Theory. Academic, New York (1972)

    Google Scholar 

  21. Kolokoltsov, V.: Markov Processes, Semigroups and Generators. De Gruyter Studies in Mathematics, vol. 38. de Gruyter, Berlin (2011).

  22. Kondo, H., Maejima, M., Sato, K.: Some properties of exponential integrals of Lévy processes and examples. Electr. Comm. Probab. 11, 291–303 (2006)

    MathSciNet  MATH  Google Scholar 

  23. Kuznetsov, A., Pardo, J.C., Savov, M.: Distributional properties of exponential functionals of Lévy processes. Electron. J. Probab. 17, 1–35 (2012)

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  25. Lévy, P.: Quelques problèmes non résolus de la théorie des fonctions caractéristiques. Ann. Mat. Pura Appl. 53(4), 315–331 (1961)

    Article  MathSciNet  Google Scholar 

  26. Liggett, T.: Continuous Time Markov Processes. An Introduction. AMS Graduate Studies in Mathematics, vol. 113. AMS, Providence (2010)

  27. Lindner, A., Maller, R.: Lévy integrals and the stationarity of generalised Ornstein-Uhlenbeck processes. Stoch. Process. Appl. 115, 1701–1722 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  28. Lindner, A., Sato, K.: Continuity properties and infinite divisibility of stationary distributions of some generalised Ornstein-Uhlenbeck processes. Ann. Probab. 37, 250–274 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  29. Loève, M.: Nouvelles classes de lois limites. Bulletin de la S.M.F. 73, 107–126 (1945)

    MATH  Google Scholar 

  30. Lucasz, E.: Characteristic Functions, 2nd edn. Griffin, London (1970)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  32. Medvegyev, P.: Stochastic Integration Theory. Oxford University Press, Oxford (2007)

    MATH  Google Scholar 

  33. Nilsen, T., Paulsen, J.: On the distribution of a randomly discounted compound Poisson process. Stoch. Process. Appl. 61, 305–310 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  34. Paulsen, J.: Risk theory in a stochastic economic environment. Stoch. Process. Appl. 46, 327–361 (2003)

    Article  MathSciNet  Google Scholar 

  35. Protter, P.E.: Stochastic Integration and Differential Equations Version 2.1, 2nd edn. Springer, Berlin (2005)

    Book  Google Scholar 

  36. Sato, K.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999)

    MATH  Google Scholar 

  37. Sato, K.: Transformations of infinitely divisible distributions via improper stochastic integrals. ALEA 3, 67–110 (2007)

    Google Scholar 

  38. Sato, K., Yamazato, M.: Operator-selfdecomposable distributions as limit distributions of processes of Ornstein-Uhlenbeck type. Stoch. Process. Appl. 17, 73–100 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  39. Schilling, R.L., Schnurr, A.: The symbol associated with the solution of a stochastic differential equation. Electron. J. Probab. 15, 1369–1393 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  40. Steutel, F.W., van Harn, K.: Infinite Divisibility of Probability Distributions on the Real Line. Marcel Dekker Inc, New York (2003)

    Book  Google Scholar 

  41. Yor, M.: Exponential Functionals of Brownian Motion and Related Processes. Springer Finance, Berlin (2001)

    Book  MATH  Google Scholar 

Download references

Acknowledgments

The authors would like to thank Makoto Maejima for fruitful discussions that initiated the investigations of this paper.

Author information

Authors and Affiliations

  1. Institut für Mathematische Statistik, Technische Universität München, Boltzmannstraße 3, 85748, Garching bei München, Germany

    Anita Behme

  2. Institut für Mathematische Stochastik, Technische Universität Braunschweig, Pockelsstr. 14, 38106, Braunschweig, Germany

    Alexander Lindner

Authors
  1. Anita Behme
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alexander Lindner
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anita Behme.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Behme, A., Lindner, A. On Exponential Functionals of Lévy Processes. J Theor Probab 28, 681–720 (2015). https://doi.org/10.1007/s10959-013-0507-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10959-013-0507-y

Keywords

Mathematics Subject Classification (2010)

Search

Navigation