IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v150y2022icp655-677.html
   My bibliography  Save this article

Strong solutions of a stochastic differential equation with irregular random drift

Author

Listed:
  • Holden, Helge
  • Karlsen, Kenneth H.
  • Pang, Peter H.C.

Abstract

We present a well-posedness result for strong solutions of one-dimensional stochastic differential equations (SDEs) of the form dX=u(ω,t,X)dt+12σ(ω,t,X)∂xσ(ω,t,X)dt+σ(ω,t,X)dW(t),where the drift coefficient u is random and irregular, with a weak derivative satisfying ∂xu=q for some q∈LωpLt∞(Lx2∩Lx1), p∈[1,∞). The random and regular noise coefficient σ may vanish. The main contribution is a pathwise uniqueness result under the assumptions that E‖q(t)−q(0)‖L2(R)2→0 as t↓0, and u satisfies the one-sided gradient bound q(ω,t,x)≤K(ω,t), where the process K(ω,t)>0 exhibits an exponential moment bound of the form Eexp(p∫tTK(s)ds)≲t−2p for small times t, for some p≥1. This study is motivated by ongoing work on the well-posedness of the stochastic Hunter–Saxton equation, a stochastic perturbation of a nonlinear transport equation that arises in the modelling of the director field of a nematic liquid crystal. In this context, the one-sided bound acts as a selection principle for dissipative weak solutions of the stochastic partial differential equation.

Suggested Citation

  • Holden, Helge & Karlsen, Kenneth H. & Pang, Peter H.C., 2022. "Strong solutions of a stochastic differential equation with irregular random drift," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 655-677.
    • Handle: RePEc:eee:spapps:v:150:y:2022:i:c:p:655-677
    DOI: 10.1016/j.spa.2022.05.006
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S030441492200120X
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2022.05.006?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. Zhang, Xicheng, 2005. "Strong solutions of SDES with singular drift and Sobolev diffusion coefficients," Stochastic Processes and their Applications, Elsevier, vol. 115(11), pages 1805-1818, November.
    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. Krylov, N.V., 2021. "On stochastic Itô processes with drift in Ld," Stochastic Processes and their Applications, Elsevier, vol. 138(C), pages 1-25.
    2. G. Prato & F. Flandoli & E. Priola & M. Röckner, 2015. "Strong Uniqueness for Stochastic Evolution Equations with Unbounded Measurable Drift Term," Journal of Theoretical Probability, Springer, vol. 28(4), pages 1571-1600, December.
    3. Wenjie Ye, 2024. "Stochastic Differential Equations with Local Growth Singular Drifts," Journal of Theoretical Probability, Springer, vol. 37(3), pages 2576-2614, September.
    4. Luo, Dejun, 2011. "Absolute continuity under flows generated by SDE with measurable drift coefficients," Stochastic Processes and their Applications, Elsevier, vol. 121(10), pages 2393-2415, October.
    5. Jinxia Wang, 2015. "Nonexplosion and Pathwise Uniqueness of Stochastic Differential Equation Driven by Continuous Semimartingale with Non-Lipschitz Coefficients," Journal of Mathematics, Hindawi, vol. 2015, pages 1-5, May.
    6. Wujun Lv & Xing Huang, 2021. "Harnack and Shift Harnack Inequalities for Degenerate (Functional) Stochastic Partial Differential Equations with Singular Drifts," Journal of Theoretical Probability, Springer, vol. 34(2), pages 827-851, June.
    7. Xian Chen & Yong Chen & Yumin Cheng & Chen Jia, 2024. "Moderate and $$L^p$$ L p Maximal Inequalities for Diffusion Processes and Conformal Martingales," Journal of Theoretical Probability, Springer, vol. 37(4), pages 2990-3014, November.
    8. Dejun Luo, 2015. "Quasi-invariance of the Stochastic Flow Associated to Itô’s SDE with Singular Time-Dependent Drift," Journal of Theoretical Probability, Springer, vol. 28(4), pages 1743-1762, December.
    9. Yang, Saisai & Zhang, Tusheng, 2023. "Strong solutions to reflecting stochastic differential equations with singular drift," Stochastic Processes and their Applications, Elsevier, vol. 156(C), pages 126-155.
    10. Xie, Longjie, 2017. "Singular SDEs with critical non-local and non-symmetric Lévy type generator," Stochastic Processes and their Applications, Elsevier, vol. 127(11), pages 3792-3824.
    11. Xia, Pengcheng & Xie, Longjie & Zhang, Xicheng & Zhao, Guohuan, 2020. "Lq(Lp)-theory of stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 130(8), pages 5188-5211.

    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:150:y:2022:i:c:p:655-677. 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.