IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v33y2020i2d10.1007_s10959-019-00967-0.html
   My bibliography  Save this article

Density Bounds for Solutions to Differential Equations Driven by Gaussian Rough Paths

Author

Listed:
  • Benjamin Gess

    (Max Planck Institute for Mathematics in the Sciences
    University of Bielefeld)

  • Cheng Ouyang

    (University of Illinois at Chicago)

  • Samy Tindel

    (Purdue University)

Abstract

We consider finite-dimensional rough differential equations driven by centered Gaussian processes. Combining Malliavin calculus, rough paths techniques and interpolation inequalities, we establish upper bounds on the density of the corresponding solution for any fixed time $$t>0$$t>0. In addition, we provide Varadhan estimates for the asymptotic behavior of the density for small noise. The emphasis is on working with general Gaussian processes with covariance function satisfying suitable abstract, checkable conditions.

Suggested Citation

  • Benjamin Gess & Cheng Ouyang & Samy Tindel, 2020. "Density Bounds for Solutions to Differential Equations Driven by Gaussian Rough Paths," Journal of Theoretical Probability, Springer, vol. 33(2), pages 611-648, June.
    • Handle: RePEc:spr:jotpro:v:33:y:2020:i:2:d:10.1007_s10959-019-00967-0
    DOI: 10.1007/s10959-019-00967-0
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10959-019-00967-0
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10959-019-00967-0?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. Lou, Shuwen & Ouyang, Cheng, 2017. "Local times of stochastic differential equations driven by fractional Brownian motions," Stochastic Processes and their Applications, Elsevier, vol. 127(11), pages 3643-3660.
    2. Baudoin, Fabrice & Ouyang, Cheng, 2011. "Small-time kernel expansion for solutions of stochastic differential equations driven by fractional Brownian motions," Stochastic Processes and their Applications, Elsevier, vol. 121(4), pages 759-792, April.
    3. Russo, Francesco & Tudor, Ciprian A., 2006. "On bifractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 116(5), pages 830-856, May.
    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. Yuzuru Inahama & Bin Pei, 2022. "Positivity of the Density for Rough Differential Equations," Journal of Theoretical Probability, Springer, vol. 35(3), pages 1863-1877, September.
    2. Song, Jian & Tindel, Samy, 2022. "Skorohod and Stratonovich integrals for controlled processes," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 569-595.

    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. Tommi Sottinen & Lauri Viitasaari, 2018. "Parameter estimation for the Langevin equation with stationary-increment Gaussian noise," Statistical Inference for Stochastic Processes, Springer, vol. 21(3), pages 569-601, October.
    2. Fan, XiLiang, 2015. "Logarithmic Sobolev inequalities for fractional diffusion," Statistics & Probability Letters, Elsevier, vol. 106(C), pages 165-172.
    3. Xiliang Fan, 2019. "Derivative Formulas and Applications for Degenerate Stochastic Differential Equations with Fractional Noises," Journal of Theoretical Probability, Springer, vol. 32(3), pages 1360-1381, September.
    4. Cristina Girolami & Giorgio Fabbri & Francesco Russo, 2014. "The covariation for Banach space valued processes and applications," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 77(1), pages 51-104, January.
    5. Harnett, Daniel & Nualart, David, 2012. "Weak convergence of the Stratonovich integral with respect to a class of Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 122(10), pages 3460-3505.
    6. Yamada, Toshihiro, 2015. "A formula of small time expansion for Young SDE driven by fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 101(C), pages 64-72.
    7. Daniel Harnett & Arturo Jaramillo & David Nualart, 2019. "Symmetric Stochastic Integrals with Respect to a Class of Self-similar Gaussian Processes," Journal of Theoretical Probability, Springer, vol. 32(3), pages 1105-1144, September.
    8. Rosanna Coviello & Cristina Di Girolami & Francesco Russo, 2011. "On stochastic calculus related to financial assets without semimartingales," Papers 1102.2050, arXiv.org.
    9. Yu, Qian & Bajja, Salwa, 2020. "Volatility estimation of general Gaussian Ornstein–Uhlenbeck process," Statistics & Probability Letters, Elsevier, vol. 163(C).
    10. Slominski, Leszek & Ziemkiewicz, Bartosz, 2009. "On weak approximations of integrals with respect to fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 79(4), pages 543-552, February.
    11. Lei, Pedro & Nualart, David, 2009. "A decomposition of the bifractional Brownian motion and some applications," Statistics & Probability Letters, Elsevier, vol. 79(5), pages 619-624, March.
    12. Skorniakov, V., 2019. "On a covariance structure of some subset of self-similar Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 129(6), pages 1903-1920.
    13. Harnett, Daniel & Nualart, David, 2018. "Central limit theorem for functionals of a generalized self-similar Gaussian process," Stochastic Processes and their Applications, Elsevier, vol. 128(2), pages 404-425.
    14. Xu Feng, 2020. "Bifractional Black-Scholes Model for Pricing European Options and Compound Options," Journal of Systems Science and Information, De Gruyter, vol. 8(4), pages 346-355, August.
    15. Baudoin, Fabrice & Ouyang, Cheng & Zhang, Xuejing, 2015. "Varadhan estimates for rough differential equations driven by fractional Brownian motions," Stochastic Processes and their Applications, Elsevier, vol. 125(2), pages 634-652.
    16. Nualart, David & Xu, Fangjun, 2019. "Asymptotic behavior for an additive functional of two independent self-similar Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 129(10), pages 3981-4008.
    17. Zhang, Xili & Xiao, Weilin, 2017. "Arbitrage with fractional Gaussian processes," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 471(C), pages 620-628.
    18. Akahori, Jiro & Song, Xiaoming & Wang, Tai-Ho, 2019. "Bridge representation and modal-path approximation," Stochastic Processes and their Applications, Elsevier, vol. 129(1), pages 174-204.
    19. Christian Bayer & Peter K. Friz & Archil Gulisashvili & Blanka Horvath & Benjamin Stemper, 2017. "Short-time near-the-money skew in rough fractional volatility models," Papers 1703.05132, arXiv.org, revised Mar 2018.

    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:spr:jotpro:v:33:y:2020:i:2:d:10.1007_s10959-019-00967-0. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.