IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v153y2022icp39-56.html
   My bibliography  Save this article

Limit theorems for local times and applications to SDEs with jumps

Author

Listed:
  • Mijatović, Aleksandar
  • Uribe Bravo, Gerónimo

Abstract

Consider a stochastic process X, regenerative at a state x which is instantaneous and regular. Let L be a regenerative local time for X at x. Suppose furthermore that X can be approximated by discrete time regenerative processes Xn for which x is accessible. We give conditions on X and Xn so that the naturally defined local time of Xn (which counts the quantity of visits to x) converges weakly to L. This limit theorem generalizes previous invariance principles that have appeared in the literature. Furthermore, it allows one to prove novel invariance principles for local times of regenerative processes converging to diffusions or to solutions of SDEs with jumps.

Suggested Citation

  • Mijatović, Aleksandar & Uribe Bravo, Gerónimo, 2022. "Limit theorems for local times and applications to SDEs with jumps," Stochastic Processes and their Applications, Elsevier, vol. 153(C), pages 39-56.
    • Handle: RePEc:eee:spapps:v:153:y:2022:i:c:p:39-56
    DOI: 10.1016/j.spa.2022.06.022
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414922001570
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2022.06.022?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. Kallenberg, Olav, 1992. "Some time change representations of stable integrals, via predictable transformations of local martingales," Stochastic Processes and their Applications, Elsevier, vol. 40(2), pages 199-223, March.
    2. Amaury Lambert & Florian Simatos, 2015. "Asymptotic Behavior of Local Times of Compound Poisson Processes with Drift in the Infinite Variance Case," Journal of Theoretical Probability, Springer, vol. 28(1), pages 41-91, March.
    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. Zhang, Xuekang & Yi, Haoran & Shu, Huisheng, 2019. "Nonparametric estimation of the trend for stochastic differential equations driven by small α-stable noises," Statistics & Probability Letters, Elsevier, vol. 151(C), pages 8-16.
    2. Shen, Leyi & Xia, Xiaoyu & Yan, Litan, 2022. "Least squares estimation for the linear self-repelling diffusion driven by α-stable motions," Statistics & Probability Letters, Elsevier, vol. 181(C).
    3. Bert Zwart, 2022. "Conjectures on symmetric queues in heavy traffic," Queueing Systems: Theory and Applications, Springer, vol. 100(3), pages 369-371, April.
    4. Matyas Barczy & Leif Doering & Zenghu Li & Gyula Pap, 2013. "Stationarity and ergodicity for an affine two factor model," Papers 1302.2534, arXiv.org, revised Sep 2013.
    5. Hu, Yaozhong & Long, Hongwei, 2009. "Least squares estimator for Ornstein-Uhlenbeck processes driven by [alpha]-stable motions," Stochastic Processes and their Applications, Elsevier, vol. 119(8), pages 2465-2480, August.
    6. Zanzotto, P. A., 1997. "On solutions of one-dimensional stochastic differential equations driven by stable Lévy motion," Stochastic Processes and their Applications, Elsevier, vol. 68(2), pages 209-228, June.
    7. Shu, Huisheng & Jiang, Ziwei & Zhang, Xuekang, 2023. "Parameter estimation for integrated Ornstein–Uhlenbeck processes with small Lévy noises," Statistics & Probability Letters, Elsevier, vol. 199(C).
    8. Barndorff-Nielsen, Ole E. & Benth, Fred Espen & Pedersen, Jan & Veraart, Almut E.D., 2014. "On stochastic integration for volatility modulated Lévy-driven Volterra processes," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 812-847.
    9. Xuekang Zhang & Huisheng Shu & Haoran Yi, 2023. "Parameter Estimation for Ornstein–Uhlenbeck Driven by Ornstein–Uhlenbeck Processes with Small Lévy Noises," Journal of Theoretical Probability, Springer, vol. 36(1), pages 78-98, March.
    10. Wiktorsson, Magnus, 2002. "Simulation of stochastic integrals with respect to Lévy processes of type G," Stochastic Processes and their Applications, Elsevier, vol. 101(1), pages 113-125, September.
    11. Long, Hongwei, 2009. "Least squares estimator for discretely observed Ornstein-Uhlenbeck processes with small Lévy noises," Statistics & Probability Letters, Elsevier, vol. 79(19), pages 2076-2085, October.

    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:153:y:2022:i:c:p:39-56. 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.