IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v32y2019i3d10.1007_s10959-018-0809-1.html
   My bibliography  Save this article

Scaling Limits of Solutions of Linear Stochastic Differential Equations Driven by Lévy White Noises

Author

Listed:
  • Julien Fageot

    (Biomedical Imaging Group)

  • Michael Unser

    (Biomedical Imaging Group)

Abstract

Consider a random process s that is a solution of the stochastic differential equation $$\mathrm {L}s = w$$ L s = w with $$\mathrm {L}$$ L a homogeneous operator and w a multidimensional Lévy white noise. In this paper, we study the asymptotic effect of zooming in or zooming out of the process s. More precisely, we give sufficient conditions on $$\mathrm {L}$$ L and w such that $$a^H s(\cdot / a)$$ a H s ( · / a ) converges in law to a non-trivial self-similar process for some H, when $$a \rightarrow 0$$ a → 0 (coarse-scale behavior) or $$a \rightarrow \infty $$ a → ∞ (fine-scale behavior). The parameter H depends on the homogeneity order of the operator $$\mathrm {L}$$ L and the Blumenthal–Getoor and Pruitt indices associated with the Lévy white noise w. Finally, we apply our general results to several famous classes of random processes and random fields and illustrate our results on simulations of Lévy processes.

Suggested Citation

  • Julien Fageot & Michael Unser, 2019. "Scaling Limits of Solutions of Linear Stochastic Differential Equations Driven by Lévy White Noises," Journal of Theoretical Probability, Springer, vol. 32(3), pages 1166-1189, September.
    • Handle: RePEc:spr:jotpro:v:32:y:2019:i:3:d:10.1007_s10959-018-0809-1
    DOI: 10.1007/s10959-018-0809-1
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10959-018-0809-1
    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/s10959-018-0809-1?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. Deng, Chang-Song & Schilling, René L., 2015. "On shift Harnack inequalities for subordinate semigroups and moment estimates for Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 125(10), pages 3851-3878.
    2. Fageot, Julien & Fallah, Alireza & Unser, Michael, 2017. "Multidimensional Lévy white noise in weighted Besov spaces," Stochastic Processes and their Applications, Elsevier, vol. 127(5), pages 1599-1621.
    3. Breton, Jean-Christophe & Dombry, Clément, 2009. "Rescaled weighted random ball models and stable self-similar random fields," Stochastic Processes and their Applications, Elsevier, vol. 119(10), pages 3633-3652, October.
    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. Fageot, Julien & Fallah, Alireza & Unser, Michael, 2017. "Multidimensional Lévy white noise in weighted Besov spaces," Stochastic Processes and their Applications, Elsevier, vol. 127(5), pages 1599-1621.
    2. Robin Merkle & Andrea Barth, 2023. "On Properties and Applications of Gaussian Subordinated Lévy Fields," Methodology and Computing in Applied Probability, Springer, vol. 25(2), pages 1-33, June.
    3. Fageot, Julien & Humeau, Thomas, 2021. "The domain of definition of the Lévy white noise," Stochastic Processes and their Applications, Elsevier, vol. 135(C), pages 75-102.
    4. Grahovac, Danijel, 2022. "Intermittency in the small-time behavior of Lévy processes," Statistics & Probability Letters, Elsevier, vol. 187(C).
    5. Julien Fageot & Michael Unser & John Paul Ward, 2020. "The $$n$$n-term Approximation of Periodic Generalized Lévy Processes," Journal of Theoretical Probability, Springer, vol. 33(1), pages 180-200, March.
    6. Song, Yan-Hong, 2016. "Algebraic ergodicity for SDEs driven by Lévy processes," Statistics & Probability Letters, Elsevier, vol. 119(C), pages 108-115.
    7. Chang-Song Deng, 2020. "Subgeometric Rates of Convergence for Discrete-Time Markov Chains Under Discrete-Time Subordination," Journal of Theoretical Probability, Springer, vol. 33(1), pages 522-532, March.
    8. Kühn, Franziska & Schilling, René L., 2019. "Strong convergence of the Euler–Maruyama approximation for a class of Lévy-driven SDEs," Stochastic Processes and their Applications, Elsevier, vol. 129(8), pages 2654-2680.
    9. Kühn, Franziska, 2017. "Existence and estimates of moments for Lévy-type processes," Stochastic Processes and their Applications, Elsevier, vol. 127(3), pages 1018-1041.
    10. Iosif Pinelis, 2018. "Positive-part moments via characteristic functions, and more general expressions," Journal of Theoretical Probability, Springer, vol. 31(1), pages 527-555, March.

    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:32:y:2019:i:3:d:10.1007_s10959-018-0809-1. 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.