IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v128y2018i8p2642-2687.html
   My bibliography  Save this article

Time inhomogeneous Stochastic Differential Equations involving the local time of the unknown process, and associated parabolic operators

Author

Listed:
  • Étoré, Pierre
  • Martinez, Miguel

Abstract

In this paper we study time inhomogeneous versions of one-dimensional Stochastic Differential Equations (SDE) involving the Local Time of the unknown process on curves. After proving existence and uniqueness for these SDEs under mild assumptions, we explore their link with Parabolic Differential Equations (PDE) with transmission conditions. We study the regularity of solutions of such PDEs and ensure the validity of a Feynman–Kac representation formula. These results are then used to characterize the solutions of these SDEs as time inhomogeneous Markov Feller processes.

Suggested Citation

  • Étoré, Pierre & Martinez, Miguel, 2018. "Time inhomogeneous Stochastic Differential Equations involving the local time of the unknown process, and associated parabolic operators," Stochastic Processes and their Applications, Elsevier, vol. 128(8), pages 2642-2687.
    • Handle: RePEc:eee:spapps:v:128:y:2018:i:8:p:2642-2687
    DOI: 10.1016/j.spa.2017.09.018
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414917302478
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2017.09.018?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. Trutnau, Gerald, 2011. "Pathwise uniqueness of the squared Bessel and CIR processes with skew reflection on a deterministic time dependent curve," Stochastic Processes and their Applications, Elsevier, vol. 121(8), pages 1845-1863, August.
    2. Blei, Stefan & Engelbert, Hans-Jürgen, 2013. "One-dimensional stochastic differential equations with generalized and singular drift," Stochastic Processes and their Applications, Elsevier, vol. 123(12), pages 4337-4372.
    3. Trutnau, Gerald, 2010. "Weak existence of the squared Bessel and CIR processes with skew reflection on a deterministic time-dependent curve," Stochastic Processes and their Applications, Elsevier, vol. 120(4), pages 381-402, April.
    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. Benabdallah Mohsine & Hiderah Kamal, 2018. "Strong rate of convergence for the Euler–Maruyama approximation of one-dimensional stochastic differential equations involving the local time at point zero," Monte Carlo Methods and Applications, De Gruyter, vol. 24(4), pages 249-262, December.
    2. Olivier Menoukeu-Pamen & Youssef Ouknine & Ludovic Tangpi, 2019. "Pathwise Uniqueness of Non-uniformly Elliptic SDEs with Rough Coefficients," Journal of Theoretical Probability, Springer, vol. 32(4), pages 1892-1908, December.
    3. Meier, Christian & Li, Lingfei & Zhang, Gongqiu, 2023. "Simulation of multidimensional diffusions with sticky boundaries via Markov chain approximation," European Journal of Operational Research, Elsevier, vol. 305(3), pages 1292-1308.

    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. Guangli Xu & Xingchun Wang, 2021. "On the Transition Density and First Hitting Time Distributions of the Doubly Skewed CIR Process," Methodology and Computing in Applied Probability, Springer, vol. 23(3), pages 735-752, September.
    2. Shiyu Song & Guangli Xu & Yongjin Wang, 2016. "On First Hitting Times for Skew CIR Processes," Methodology and Computing in Applied Probability, Springer, vol. 18(1), pages 169-180, March.
    3. Tian, Yingxu & Zhang, Haoyan, 2018. "Skew CIR process, conditional characteristic function, moments and bond pricing," Applied Mathematics and Computation, Elsevier, vol. 329(C), pages 230-238.
    4. Xu, Guangli & Wang, Yongjin, 2016. "On stability of the Markov-modulated skew CIR process," Statistics & Probability Letters, Elsevier, vol. 109(C), pages 139-144.
    5. Guangli Xu & Shiyu Song & Yongjin Wang, 2016. "The Valuation Of Options On Foreign Exchange Rate In A Target Zone," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 19(03), pages 1-19, May.
    6. Pajor-Gyulai, Zs. & Salins, M., 2017. "On dynamical systems perturbed by a null-recurrent motion: The general case," Stochastic Processes and their Applications, Elsevier, vol. 127(6), pages 1960-1997.
    7. Bachmann, Stefan, 2020. "On the strong Feller property for stochastic delay differential equations with singular drift," Stochastic Processes and their Applications, Elsevier, vol. 130(8), pages 4563-4592.
    8. Benabdallah Mohsine & Hiderah Kamal, 2018. "Strong rate of convergence for the Euler–Maruyama approximation of one-dimensional stochastic differential equations involving the local time at point zero," Monte Carlo Methods and Applications, De Gruyter, vol. 24(4), pages 249-262, December.
    9. Trutnau, Gerald, 2011. "Pathwise uniqueness of the squared Bessel and CIR processes with skew reflection on a deterministic time dependent curve," Stochastic Processes and their Applications, Elsevier, vol. 121(8), pages 1845-1863, August.

    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:128:y:2018:i:8:p:2642-2687. 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.