IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v127y2017i7p2281-2315.html
   My bibliography  Save this article

Stochastic differential equation for Brox diffusion

Author

Listed:
  • Hu, Yaozhong
  • Lê, Khoa
  • Mytnik, Leonid

Abstract

This paper studies the weak and strong solutions to the stochastic differential equation dX(t)=−12Ẇ(X(t))dt+dB(t), where (B(t),t≥0) is a standard Brownian motion and W(x) is a two sided Brownian motion, independent of B. It is shown that the Itô–McKean representation associated with any Brownian motion (independent of W) is a weak solution to the above equation. It is also shown that there exists a unique strong solution to the equation. Itô calculus for the solution is developed. For dealing with the singularity of drift term ∫0TẆ(X(t))dt, the main idea is to use the concept of local time together with the polygonal approximation Wπ. Some new results on the local time of Brownian motion needed in our proof are established.

Suggested Citation

  • Hu, Yaozhong & Lê, Khoa & Mytnik, Leonid, 2017. "Stochastic differential equation for Brox diffusion," Stochastic Processes and their Applications, Elsevier, vol. 127(7), pages 2281-2315.
    • Handle: RePEc:eee:spapps:v:127:y:2017:i:7:p:2281-2315
    DOI: 10.1016/j.spa.2016.10.010
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414916301983
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2016.10.010?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. Hu, Yaozhong & Le, Khoa, 2013. "A multiparameter Garsia–Rodemich–Rumsey inequality and some applications," Stochastic Processes and their Applications, Elsevier, vol. 123(9), pages 3359-3377.
    2. Diel, Roland, 2011. "Almost sure asymptotics for the local time of a diffusion in Brownian environment," Stochastic Processes and their Applications, Elsevier, vol. 121(10), pages 2303-2330, October.
    3. Shi, Zhan, 1998. "A local time curiosity in random environment," Stochastic Processes and their Applications, Elsevier, vol. 76(2), pages 231-250, August.
    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. Ghazinoory, Sepehr & Aghaei, Parvaneh, 2021. "Differences between policy assessment & policy evaluation; a case study on supportive policies for knowledge-based firms," Technological Forecasting and Social Change, Elsevier, vol. 169(C).
    2. De Angelis, Tiziano & Germain, Maximilien & Issoglio, Elena, 2022. "A numerical scheme for stochastic differential equations with distributional drift," Stochastic Processes and their Applications, Elsevier, vol. 154(C), pages 55-90.

    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. Grégoire Véchambre, 2023. "Almost Sure Behavior for the Local Time of a Diffusion in a Spectrally Negative Lévy Environment," Journal of Theoretical Probability, Springer, vol. 36(2), pages 876-925, June.
    2. Andreoletti, Pierre, 2006. "On the concentration of Sinai's walk," Stochastic Processes and their Applications, Elsevier, vol. 116(10), pages 1377-1408, October.
    3. Hu, Yueyun, 2000. "Tightness of localization and return time in random environment," Stochastic Processes and their Applications, Elsevier, vol. 86(1), pages 81-101, March.
    4. Gutierrez-Pavón, Jonathan & Pacheco, Carlos G., 2022. "Quenched distributions for the maximum, minimum and local time of the Brox diffusion," Statistics & Probability Letters, Elsevier, vol. 180(C).
    5. Andreoletti, Pierre, 2007. "Almost sure estimates for the concentration neighborhood of Sinai's walk," Stochastic Processes and their Applications, Elsevier, vol. 117(10), pages 1473-1490, October.
    6. Gutierrez-Pavón, Jonathan & Pacheco, Carlos G., 2020. "A density for the local time of the Brox diffusion," Statistics & Probability Letters, Elsevier, vol. 163(C).
    7. Gantert, Nina & Shi, Zhan, 2002. "Many visits to a single site by a transient random walk in random environment," Stochastic Processes and their Applications, Elsevier, vol. 99(2), pages 159-176, June.
    8. Pierre Andreoletti & Roland Diel, 2011. "Limit Law of the Local Time for Brox’s Diffusion," Journal of Theoretical Probability, Springer, vol. 24(3), pages 634-656, September.
    9. Zindy, Olivier, 2008. "Upper limits of Sinai's walk in random scenery," Stochastic Processes and their Applications, Elsevier, vol. 118(6), pages 981-1003, June.
    10. Diel, Roland, 2011. "Almost sure asymptotics for the local time of a diffusion in Brownian environment," Stochastic Processes and their Applications, Elsevier, vol. 121(10), pages 2303-2330, 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:127:y:2017:i:7:p:2281-2315. 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.