Skip to main content
SpringerLink
Log in

A Functional Non-Central Limit Theorem for Jump-Diffusions with Periodic Coefficients Driven by Stable Lévy-Noise

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

Abstract

We prove a functional non-central limit theorem for jump-diffusions with periodic coefficients driven by stable Lévy-processes with stability index α>1. The limit process turns out to be an α-stable Lévy process with an averaged jump-measure. Unlike in the situation where the diffusion is driven by Brownian motion, there is no drift related enhancement of diffusivity.

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

    MATH  Google Scholar 

  2. Bensoussan, A., Lions, J.L., Papanicolaou, G.: Asymptotic Analysis for Periodic Structures. North-Holland, Amsterdam (1978)

    MATH  Google Scholar 

  3. Bhattacharya, R.: A central limit theorem for diffusions with periodic coefficients. Ann. Probab. 13, 385–396 (1985)

    MATH  MathSciNet  Google Scholar 

  4. Doob, J.L.: Stochastic Processes. Wiley, New York (1953)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  6. Franke, B.: The scaling limit behaviour of periodic stable-like processes. Bernoulli 12, 551–570 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  7. Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes, 2nd edn. North-Holland, Amsterdam (1989)

    MATH  Google Scholar 

  8. Jacob, N.: Pseudo-Differential Operators and Markov Processes, vol. 1. Imperial College Press (2001)

  9. Jacod, J., Shiryaev, A.: Limit Theorems for Stochastic Processes. Grundlehren der mathematischen Wissenschaften, vol. 288. Springer, Berlin (1987)

    MATH  Google Scholar 

  10. Kolokoltsov, V.: Symmetric stable laws and stable-like jump diffusions. Proc. Lond. Math. Soc. 80, 725–768 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  11. Kolokoltsov, V.: Semiclassical Analysis for Diffusions and Stochastic Processes. Lecture Notes in Mathematics, vol. 1724. Springer, Berlin (2000)

    MATH  Google Scholar 

  12. Picard, J., Savona, C.: Smoothness of harmonic functions for processes with jumps. Stoch. Process. Appl. 87, 69–91 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  13. Protter, P.: Stochastic Integration and Differential Equations, 2nd edn. Applications of Mathematics, vol. 21. Springer, Berlin (2004)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fakultät für Mathematik, Ruhr Universität Bochum, Universitätsstr. 150, 44780, Bochum, Germany

    Brice Franke

Authors
  1. Brice Franke
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Brice Franke.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Franke, B. A Functional Non-Central Limit Theorem for Jump-Diffusions with Periodic Coefficients Driven by Stable Lévy-Noise. J Theor Probab 20, 1087–1100 (2007). https://doi.org/10.1007/s10959-007-0099-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10959-007-0099-5

Keywords

Search

Navigation