IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v130y2020i5p3053-3094.html
   My bibliography  Save this article

Gradient estimates and ergodicity for SDEs driven by multiplicative Lévy noises via coupling

Author

Listed:
  • Liang, Mingjie
  • Wang, Jian

Abstract

We consider SDEs driven by multiplicative pure jump Lévy noises, where Lévy processes are not necessarily comparable to α-stable-like processes. By assuming that the SDE has a unique strong solution, we obtain gradient estimates of the associated semigroup when the drift term is locally Hölder continuous, and we establish the ergodicity of the process both in the L1-Wasserstein distance and the total variation, when the coefficients are dissipative for large distances. The proof is based on a new explicit Markov coupling for SDEs driven by multiplicative pure jump Lévy noises, which has been open for a long time in this area.

Suggested Citation

  • Liang, Mingjie & Wang, Jian, 2020. "Gradient estimates and ergodicity for SDEs driven by multiplicative Lévy noises via coupling," Stochastic Processes and their Applications, Elsevier, vol. 130(5), pages 3053-3094.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:5:p:3053-3094
    DOI: 10.1016/j.spa.2019.09.001
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414918307646
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2019.09.001?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. Bass, Richard F. & Burdzy, Krzysztof & Chen, Zhen-Qing, 2004. "Stochastic differential equations driven by stable processes for which pathwise uniqueness fails," Stochastic Processes and their Applications, Elsevier, vol. 111(1), pages 1-15, May.
    2. Majka, Mateusz B., 2017. "Coupling and exponential ergodicity for stochastic differential equations driven by Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 127(12), pages 4083-4125.
    3. Mátyás Barczy & Zenghu Li & Gyula Pap, 2015. "Yamada-Watanabe Results for Stochastic Differential Equations with Jumps," International Journal of Stochastic Analysis, Hindawi, vol. 2015, pages 1-23, January.
    4. Wang, Linlin & Xie, Longjie & Zhang, Xicheng, 2015. "Derivative formulae for SDEs driven by multiplicative α-stable-like processes," Stochastic Processes and their Applications, Elsevier, vol. 125(3), pages 867-885.
    5. Luo, Dejun & Wang, Jian, 2019. "Refined basic couplings and Wasserstein-type distances for SDEs with Lévy noises," Stochastic Processes and their Applications, Elsevier, vol. 129(9), pages 3129-3173.
    6. Kulik, Alexey M., 2009. "Exponential ergodicity of the solutions to SDE's with a jump noise," Stochastic Processes and their Applications, Elsevier, vol. 119(2), pages 602-632, February.
    7. Masuda, Hiroki, 2007. "Ergodicity and exponential [beta]-mixing bounds for multidimensional diffusions with jumps," Stochastic Processes and their Applications, Elsevier, vol. 117(1), pages 35-56, January.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Kulczycki, Tadeusz & Ryznar, Michał, 2020. "Semigroup properties of solutions of SDEs driven by Lévy processes with independent coordinates," Stochastic Processes and their Applications, Elsevier, vol. 130(12), pages 7185-7217.
    2. Bao, Jianhai & Wang, Jian, 2022. "Coupling approach for exponential ergodicity of stochastic Hamiltonian systems with Lévy noises," Stochastic Processes and their Applications, Elsevier, vol. 146(C), pages 114-142.
    3. Huang, Lu-Jing & Majka, Mateusz B. & Wang, Jian, 2022. "Strict Kantorovich contractions for Markov chains and Euler schemes with general noise," Stochastic Processes and their Applications, Elsevier, vol. 151(C), pages 307-341.

    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. Bao, Jianhai & Wang, Jian, 2022. "Coupling approach for exponential ergodicity of stochastic Hamiltonian systems with Lévy noises," Stochastic Processes and their Applications, Elsevier, vol. 146(C), pages 114-142.
    2. E. Löcherbach, 2020. "Convergence to Equilibrium for Time-Inhomogeneous Jump Diffusions with State-Dependent Jump Intensity," Journal of Theoretical Probability, Springer, vol. 33(4), pages 2280-2314, December.
    3. Huang, Lu-Jing & Majka, Mateusz B. & Wang, Jian, 2022. "Strict Kantorovich contractions for Markov chains and Euler schemes with general noise," Stochastic Processes and their Applications, Elsevier, vol. 151(C), pages 307-341.
    4. Palczewski, Jan & Stettner, Łukasz, 2014. "Infinite horizon stopping problems with (nearly) total reward criteria," Stochastic Processes and their Applications, Elsevier, vol. 124(12), pages 3887-3920.
    5. Luo, Dejun & Wang, Jian, 2019. "Refined basic couplings and Wasserstein-type distances for SDEs with Lévy noises," Stochastic Processes and their Applications, Elsevier, vol. 129(9), pages 3129-3173.
    6. Jianhai Bao & Jian Wang, 2023. "Coupling methods and exponential ergodicity for two‐factor affine processes," Mathematische Nachrichten, Wiley Blackwell, vol. 296(5), pages 1716-1736, May.
    7. Oleksii Kulyk, 2023. "Support Theorem for Lévy-driven Stochastic Differential Equations," Journal of Theoretical Probability, Springer, vol. 36(3), pages 1720-1742, September.
    8. Schmisser, Émeline, 2019. "Non parametric estimation of the diffusion coefficients of a diffusion with jumps," Stochastic Processes and their Applications, Elsevier, vol. 129(12), pages 5364-5405.
    9. Fu, Zongfei & Li, Zenghu, 2010. "Stochastic equations of non-negative processes with jumps," Stochastic Processes and their Applications, Elsevier, vol. 120(3), pages 306-330, March.
    10. Noh, Jungsik & Lee, Seung Y. & Lee, Sangyeol, 2012. "Quantile regression estimation for discretely observed SDE models with compound Poisson jumps," Economics Letters, Elsevier, vol. 117(3), pages 734-738.
    11. Yuma Uehara, 2023. "Bootstrap method for misspecified ergodic Lévy driven stochastic differential equation models," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 75(4), pages 533-565, August.
    12. Yulin Song, 2020. "Gradient Estimates and Exponential Ergodicity for Mean-Field SDEs with Jumps," Journal of Theoretical Probability, Springer, vol. 33(1), pages 201-238, March.
    13. Ascione, Giacomo & Mehrdoust, Farshid & Orlando, Giuseppe & Samimi, Oldouz, 2023. "Foreign Exchange Options on Heston-CIR Model Under Lévy Process Framework," Applied Mathematics and Computation, Elsevier, vol. 446(C).
    14. Wang, Tao, 2022. "Ergodic convergence rates for time-changed symmetric Lévy processes in dimension one," Statistics & Probability Letters, Elsevier, vol. 183(C).
    15. Valentin Courgeau & Almut E. D. Veraart, 2022. "Likelihood theory for the graph Ornstein-Uhlenbeck process," Statistical Inference for Stochastic Processes, Springer, vol. 25(2), pages 227-260, July.
    16. Jakobsen, Nina Munkholt & Sørensen, Michael, 2019. "Estimating functions for jump–diffusions," Stochastic Processes and their Applications, Elsevier, vol. 129(9), pages 3282-3318.
    17. Sun, Xiaobin & Xie, Longjie & Xie, Yingchao, 2020. "Derivative formula for the Feynman–Kac semigroup of SDEs driven by rotationally invariant α-stable process," Statistics & Probability Letters, Elsevier, vol. 158(C).
    18. Song, Yan-Hong, 2016. "Algebraic ergodicity for SDEs driven by Lévy processes," Statistics & Probability Letters, Elsevier, vol. 119(C), pages 108-115.
    19. Shukai Chen & Rongjuan Fang & Xiangqi Zheng, 2023. "Wasserstein-Type Distances of Two-Type Continuous-State Branching Processes in Lévy Random Environments," Journal of Theoretical Probability, Springer, vol. 36(3), pages 1572-1590, September.
    20. Mátyás Barczy & Zenghu Li & Gyula Pap, 2016. "Moment Formulas for Multitype Continuous State and Continuous Time Branching Process with Immigration," Journal of Theoretical Probability, Springer, vol. 29(3), pages 958-995, September.

    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:130:y:2020:i:5:p:3053-3094. 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.