IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v130y2020i5p2639-2674.html
   My bibliography  Save this article

On the support of solutions to stochastic differential equations with path-dependent coefficients

Author

Listed:
  • Cont, Rama
  • Kalinin, Alexander

Abstract

Given a stochastic differential equation with path-dependent coefficients driven by a multidimensional Wiener process, we show that the support of the law of the solution is given by the image of the Cameron–Martin space under the flow of mild solutions to a system of path-dependent ordinary differential equations. Our result extends the Stroock–Varadhan support theorem for diffusion processes to the case of SDEs with path-dependent coefficients. The proof is based on functional Itô calculus.

Suggested Citation

  • Cont, Rama & Kalinin, Alexander, 2020. "On the support of solutions to stochastic differential equations with path-dependent coefficients," Stochastic Processes and their Applications, Elsevier, vol. 130(5), pages 2639-2674.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:5:p:2639-2674
    DOI: 10.1016/j.spa.2019.07.015
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414918305568
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2019.07.015?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. Ledoux, M. & Qian, Z. & Zhang, T., 2002. "Large deviations and support theorem for diffusion processes via rough paths," Stochastic Processes and their Applications, Elsevier, vol. 102(2), pages 265-283, December.
    2. Bruno Dupire, 2019. "Functional Itô calculus," Quantitative Finance, Taylor & Francis Journals, vol. 19(5), pages 721-729, May.
    3. Cont, Rama & Lu, Yi, 2016. "Weak approximation of martingale representations," Stochastic Processes and their Applications, Elsevier, vol. 126(3), pages 857-882.
    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. Damiano Brigo & Federico Graceffa & Alexander Kalinin, 2021. "Mild to classical solutions for XVA equations under stochastic volatility," Papers 2112.11808, arXiv.org.

    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. Zhou Fang, 2023. "Continuous-Time Path-Dependent Exploratory Mean-Variance Portfolio Construction," Papers 2303.02298, arXiv.org.
    2. Ofelia Bonesini & Antoine Jacquier & Alexandre Pannier, 2023. "Rough volatility, path-dependent PDEs and weak rates of convergence," Papers 2304.03042, arXiv.org.
    3. Gautier, Eric, 2005. "Uniform large deviations for the nonlinear Schrodinger equation with multiplicative noise," Stochastic Processes and their Applications, Elsevier, vol. 115(12), pages 1904-1927, December.
    4. Bardina, X. & Nourdin, I. & Rovira, C. & Tindel, S., 2010. "Weak approximation of a fractional SDE," Stochastic Processes and their Applications, Elsevier, vol. 120(1), pages 39-65, January.
    5. Christian Bayer & Paul Hager & Sebastian Riedel & John Schoenmakers, 2021. "Optimal stopping with signatures," Papers 2105.00778, arXiv.org.
    6. Dmitry Kramkov & Sergio Pulido, 2016. "Stability and analytic expansions of local solutions of systems of quadratic BSDEs with applications to a price impact model," Post-Print hal-01181147, HAL.
    7. Andrew L. Allan & Chong Liu & David J. Promel, 2021. "A C\`adl\`ag Rough Path Foundation for Robust Finance," Papers 2109.04225, arXiv.org, revised May 2023.
    8. Henry Chiu & Rama Cont, 2023. "A model‐free approach to continuous‐time finance," Mathematical Finance, Wiley Blackwell, vol. 33(2), pages 257-273, April.
    9. Georgii Riabov & Aleh Tsyvinski, 2021. "Policy with stochastic hysteresis," Papers 2104.10225, arXiv.org.
    10. Nam, Kihun, 2021. "Locally Lipschitz BSDE driven by a continuous martingale a path-derivative approach," Stochastic Processes and their Applications, Elsevier, vol. 141(C), pages 376-411.
    11. Anton Plaksin, 2020. "Minimax and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations for Time-Delay Systems," Journal of Optimization Theory and Applications, Springer, vol. 187(1), pages 22-42, October.
    12. Bruno Bouchard & Xiaolu Tan, 2021. "A quasi-sure optional decomposition and super-hedging result on the Skorokhod space," Finance and Stochastics, Springer, vol. 25(3), pages 505-528, July.
    13. Li, Xiaoyue & Mao, Xuerong & Song, Guoting, 2024. "An explicit approximation for super-linear stochastic functional differential equations," Stochastic Processes and their Applications, Elsevier, vol. 169(C).
    14. Marc Sabate-Vidales & David v{S}iv{s}ka & Lukasz Szpruch, 2020. "Solving path dependent PDEs with LSTM networks and path signatures," Papers 2011.10630, arXiv.org.
    15. Shreya Bose & Ibrahim Ekren, 2021. "Multidimensional Kyle-Back model with a risk averse informed trader," Papers 2111.01957, arXiv.org.
    16. Christa Cuchiero & Janka Moller, 2023. "Signature Methods in Stochastic Portfolio Theory," Papers 2310.02322, arXiv.org, revised Oct 2024.
    17. Qi Feng & Man Luo & Zhaoyu Zhang, 2021. "Deep Signature FBSDE Algorithm," Papers 2108.10504, arXiv.org, revised Aug 2022.
    18. Blanka Horvath & Josef Teichmann & Zan Zuric, 2021. "Deep Hedging under Rough Volatility," Papers 2102.01962, arXiv.org.
    19. Shigeki Aida, 2024. "Rough Differential Equations Containing Path-Dependent Bounded Variation Terms," Journal of Theoretical Probability, Springer, vol. 37(3), pages 2130-2183, September.
    20. Brian Huge & Antoine Savine, 2020. "Differential Machine Learning," Papers 2005.02347, arXiv.org, revised Sep 2020.

    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:2639-2674. 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.