IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v28y2015i1d10.1007_s10959-012-0476-6.html
   My bibliography  Save this article

Inversions of Lévy Measures and the Relation Between Long and Short Time Behavior of Lévy Processes

Author

Listed:
  • Michael Grabchak

    (The University of North Carolina at Charlotte)

Abstract

The inversion of a Lévy measure was first introduced (under a different name) in Sato (ALEA Lat Am J Probab Math Stat 3:67–110, 2007). We generalize the definition and give some properties. We then use inversions to derive a relationship between weak convergence of a Lévy process to an infinite variance stable distribution when time approaches zero and weak convergence of a different Lévy process as time approaches infinity. This allows us to get self-contained conditions for a Lévy process to converge to an infinite variance stable distribution as time approaches zero. We formulate our results both for general Lévy processes and for the important class of tempered stable Lévy processes. For this latter class, we give detailed results in terms of their Rosiński measures.

Suggested Citation

  • Michael Grabchak, 2015. "Inversions of Lévy Measures and the Relation Between Long and Short Time Behavior of Lévy Processes," Journal of Theoretical Probability, Springer, vol. 28(1), pages 184-197, March.
    • Handle: RePEc:spr:jotpro:v:28:y:2015:i:1:d:10.1007_s10959-012-0476-6
    DOI: 10.1007/s10959-012-0476-6
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10959-012-0476-6
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10959-012-0476-6?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Michael Grabchak & Gennady Samorodnitsky, 2010. "Do financial returns have finite or infinite variance? A paradox and an explanation," Quantitative Finance, Taylor & Francis Journals, vol. 10(8), pages 883-893.
    2. Jurek, Zbigniew J., 2007. "Random integral representations for free-infinitely divisible and tempered stable distributions," Statistics & Probability Letters, Elsevier, vol. 77(4), pages 417-425, February.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Jurek, Zbigniew J., 2018. "Remarks on compositions of some random integral mappings," Statistics & Probability Letters, Elsevier, vol. 137(C), pages 277-282.
    2. Yunfei Xia & Michael Grabchak, 2024. "Pricing multi-asset options with tempered stable distributions," Financial Innovation, Springer;Southwestern University of Finance and Economics, vol. 10(1), pages 1-24, December.
    3. Li, Hengxin & Wang, Ruodu, 2023. "PELVE: Probability Equivalent Level of VaR and ES," Journal of Econometrics, Elsevier, vol. 234(1), pages 353-370.
    4. Wied, Dominik & Dehling, Herold & van Kampen, Maarten & Vogel, Daniel, 2014. "A fluctuation test for constant Spearman’s rho with nuisance-free limit distribution," Computational Statistics & Data Analysis, Elsevier, vol. 76(C), pages 723-736.
    5. Borak, Szymon & Misiorek, Adam & Weron, Rafał, 2010. "Models for heavy-tailed asset returns," SFB 649 Discussion Papers 2010-049, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    6. Michael Grabchak, 2021. "On the transition laws of p-tempered $$\alpha $$ α -stable OU-processes," Computational Statistics, Springer, vol. 36(2), pages 1415-1436, June.
    7. A. H. Nzokem & V. T. Montshiwa, 2022. "Fitting Generalized Tempered Stable distribution: Fractional Fourier Transform (FRFT) Approach," Papers 2205.00586, arXiv.org, revised Jun 2022.
    8. Stoyanov, Stoyan V. & Rachev, Svetlozar T. & Fabozzi, Frank J., 2013. "CVaR sensitivity with respect to tail thickness," Journal of Banking & Finance, Elsevier, vol. 37(3), pages 977-988.
    9. Aubain Nzokem & Daniel Maposa, 2024. "Fitting the Seven-Parameter Generalized Tempered Stable Distribution to Financial Data," JRFM, MDPI, vol. 17(12), pages 1-29, November.
    10. Masuda, Hiroki, 2019. "Non-Gaussian quasi-likelihood estimation of SDE driven by locally stable Lévy process," Stochastic Processes and their Applications, Elsevier, vol. 129(3), pages 1013-1059.
    11. Aleksy Leeuwenkamp & Wentao Hu, 2023. "New general dependence measures: construction, estimation and application to high-frequency stock returns," Papers 2309.00025, arXiv.org.
    12. Naaman, Michael & Sickles, Robin, 2015. "The Volcano Distribution with an Application to Stock Market Returns," Working Papers 15-020, Rice University, Department of Economics.
    13. De Domenico, Federica & Livan, Giacomo & Montagna, Guido & Nicrosini, Oreste, 2023. "Modeling and simulation of financial returns under non-Gaussian distributions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 622(C).
    14. Wesselhöfft, Niels & Härdle, Wolfgang Karl, 2019. "Estimating low sampling frequency risk measure by high-frequency data," IRTG 1792 Discussion Papers 2019-003, Humboldt University of Berlin, International Research Training Group 1792 "High Dimensional Nonstationary Time Series".
    15. Molina-Muñoz, Jesús & Mora-Valencia, Andrés & Perote, Javier, 2020. "Market-crash forecasting based on the dynamics of the alpha-stable distribution," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 557(C).
    16. Aubain Nzokem & Daniel Maposa, 2024. "Fitting the seven-parameter Generalized Tempered Stable distribution to the financial data," Papers 2410.19751, arXiv.org, revised Jan 2025.
    17. Michele Bianchi & Frank Fabozzi, 2014. "Discussion of ‘on simulation and properties of the stable law’ by Devroye and James," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 23(3), pages 353-357, August.
    18. Greg Hannsgen & Tai Young-Taft, 2015. "Inside Money in a Kaldor-Kalecki-Steindl Fiscal Policy Model: The Unit of Account, Inflation, Leverage, and Financial Fragility," Economics Working Paper Archive wp_839, Levy Economics Institute.
    19. Fabozzi Frank J. & Stoyanov Stoyan V. & Rachev Svetlozar T., 2013. "Computational aspects of portfolio risk estimation in volatile markets: a survey," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 17(1), pages 103-120, February.
    20. Federica De Domenico & Giacomo Livan & Guido Montagna & Oreste Nicrosini, 2023. "Modeling and Simulation of Financial Returns under Non-Gaussian Distributions," Papers 2302.02769, arXiv.org.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:jotpro:v:28:y:2015:i:1:d:10.1007_s10959-012-0476-6. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.