IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v163y2023icp136-167.html
   My bibliography  Save this article

Approximation of the invariant measure of stable SDEs by an Euler–Maruyama scheme

Author

Listed:
  • Chen, Peng
  • Deng, Chang-Song
  • Schilling, René L.
  • Xu, Lihu

Abstract

We propose two Euler–Maruyama (EM) type numerical schemes in order to approximate the invariant measure of a stochastic differential equation (SDE) driven by an α-stable Lévy process (1<α<2): an approximation scheme with the α-stable distributed noise and a further scheme with Pareto-distributed noise. Using a discrete version of Duhamel’s principle and Bismut’s formula in Malliavin calculus, we prove that the error bounds in Wasserstein-1 distance are in the order of η1−ϵ and η2α−1, respectively, where ϵ∈(0,1) is arbitrary and η is the step size of the approximation schemes. For the Pareto-driven scheme, an explicit calculation for Ornstein–Uhlenbeck α-stable process shows that the rate η2α−1 cannot be improved.

Suggested Citation

  • Chen, Peng & Deng, Chang-Song & Schilling, René L. & Xu, Lihu, 2023. "Approximation of the invariant measure of stable SDEs by an Euler–Maruyama scheme," Stochastic Processes and their Applications, Elsevier, vol. 163(C), pages 136-167.
    • Handle: RePEc:eee:spapps:v:163:y:2023:i:c:p:136-167
    DOI: 10.1016/j.spa.2023.06.001
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414923001229
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2023.06.001?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. Lemaire, Vincent, 2007. "An adaptive scheme for the approximation of dissipative systems," Stochastic Processes and their Applications, Elsevier, vol. 117(10), pages 1491-1518, October.
    2. Aleksander Janicki & Zbigniew Michna & Aleksander Weron, 1996. "Approximation of stochastic differential equations driven by alpha-stable Levy motion," HSC Research Reports HSC/96/02, Hugo Steinhaus Center, Wroclaw University of Technology.
    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. Laruelle Sophie & Pagès Gilles, 2012. "Stochastic approximation with averaging innovation applied to Finance," Monte Carlo Methods and Applications, De Gruyter, vol. 18(1), pages 1-51, January.
    2. Panloup, Fabien, 2008. "Computation of the invariant measure for a Lévy driven SDE: Rate of convergence," Stochastic Processes and their Applications, Elsevier, vol. 118(8), pages 1351-1384, August.
    3. Cohen, Serge & Panloup, Fabien, 2011. "Approximation of stationary solutions of Gaussian driven stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 121(12), pages 2776-2801.
    4. Pagès, Gilles & Panloup, Fabien, 2014. "A mixed-step algorithm for the approximation of the stationary regime of a diffusion," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 522-565.
    5. Gilles Pagès & Clément Rey, 2023. "Discretization of the Ergodic Functional Central Limit Theorem," Journal of Theoretical Probability, Springer, vol. 36(1), pages 1-44, March.
    6. Cohen, Serge & Panloup, Fabien & Tindel, Samy, 2014. "Approximation of stationary solutions to SDEs driven by multiplicative fractional noise," Stochastic Processes and their Applications, Elsevier, vol. 124(3), pages 1197-1225.
    7. Gadat, Sébastien & Panloup, Fabien & Saadane, Sofiane, 2016. "Stochastic Heavy Ball," TSE Working Papers 16-712, Toulouse School of Economics (TSE).
    8. Valentin Konakov & Stéphane Menozzi, 2011. "Weak Error for Stable Driven Stochastic Differential Equations: Expansion of the Densities," Journal of Theoretical Probability, Springer, vol. 24(2), pages 454-478, June.
    9. Pagès Gilles & Rey Clément, 2019. "Recursive computation of the invariant distributions of Feller processes: Revisited examples and new applications," Monte Carlo Methods and Applications, De Gruyter, vol. 25(1), pages 1-36, March.
    10. Pagès, Gilles & Rey, Clément, 2020. "Recursive computation of invariant distributions of Feller processes," Stochastic Processes and their Applications, Elsevier, vol. 130(1), pages 328-365.

    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:eee:spapps:v:163:y:2023:i:c:p:136-167. 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: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    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.