IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v126y2016i8p2323-2366.html
   My bibliography  Save this article

Averaging along irregular curves and regularisation of ODEs

Author

Listed:
  • Catellier, R.
  • Gubinelli, M.

Abstract

We consider the ordinary differential equation (ODE) dxt=b(t,xt)dt+dwt where w is a continuous driving function and b is a time-dependent vector field which possibly is only a distribution in the space variable. We quantify the regularising properties of an arbitrary continuous path w on the existence and uniqueness of solutions to this equation. In this context we introduce the notion of ρ-irregularity and show that it plays a key role in some instances of the regularisation by noise phenomenon. In the particular case of a function w sampled according to the law of the fractional Brownian motion of Hurst index H∈(0,1), we prove that almost surely the ODE admits a solution for all b in the Besov–Hölder space B∞,∞α+1 with α>−1/2H. If α>1−1/2H then the solution is unique among a natural set of continuous solutions. If H>1/3 and α>3/2−1/2H or if α>2−1/2H then the equation admits a unique Lipschitz flow. Note that when α<0 the vector field b is only a distribution, nonetheless there exists a natural notion of solution for which the above results apply.

Suggested Citation

  • Catellier, R. & Gubinelli, M., 2016. "Averaging along irregular curves and regularisation of ODEs," Stochastic Processes and their Applications, Elsevier, vol. 126(8), pages 2323-2366.
    • Handle: RePEc:eee:spapps:v:126:y:2016:i:8:p:2323-2366
    DOI: 10.1016/j.spa.2016.02.002
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414916000235
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2016.02.002?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. Nualart, David & Ouknine, Youssef, 2002. "Regularization of differential equations by fractional noise," Stochastic Processes and their Applications, Elsevier, vol. 102(1), pages 103-116, November.
    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. Iksanov, Alexander & Pilipenko, Andrey, 2023. "On a skew stable Lévy process," Stochastic Processes and their Applications, Elsevier, vol. 156(C), pages 44-68.
    2. Altmeyer, Randolf & Chorowski, Jakub, 2018. "Estimation error for occupation time functionals of stationary Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 128(6), pages 1830-1848.
    3. Coffie, Emmanuel & Duedahl, Sindre & Proske, Frank, 2023. "Sensitivity analysis with respect to a stochastic stock price model with rough volatility via a Bismut–Elworthy–Li formula for singular SDEs," Stochastic Processes and their Applications, Elsevier, vol. 156(C), pages 156-195.
    4. David Baños & Salvador Ortiz-Latorre & Andrey Pilipenko & Frank Proske, 2022. "Strong Solutions of Stochastic Differential Equations with Generalized Drift and Multidimensional Fractional Brownian Initial Noise," Journal of Theoretical Probability, Springer, vol. 35(2), pages 714-771, June.
    5. Fabian A. Harang & Chengcheng Ling, 2022. "Regularity of Local Times Associated with Volterra–Lévy Processes and Path-Wise Regularization of Stochastic Differential Equations," Journal of Theoretical Probability, Springer, vol. 35(3), pages 1706-1735, September.
    6. C. Bellingeri & A. Djurdjevac & P. K. Friz & N. Tapia, 2021. "Transport and continuity equations with (very) rough noise," Partial Differential Equations and Applications, Springer, vol. 2(4), pages 1-26, August.
    7. Harang, Fabian A. & Mayorcas, Avi, 2023. "Pathwise regularisation of singular interacting particle systems and their mean field limits," Stochastic Processes and their Applications, Elsevier, vol. 159(C), pages 499-540.
    8. Gassiat, Paul & Mądry, Łukasz, 2023. "Perturbations of singular fractional SDEs," Stochastic Processes and their Applications, Elsevier, vol. 161(C), pages 137-172.
    9. Bechtold, Florian & Hofmanová, Martina, 2023. "Weak solutions for singular multiplicative SDEs via regularization by noise," Stochastic Processes and their Applications, Elsevier, vol. 157(C), pages 413-435.
    10. Altmeyer, Randolf, 2023. "Central limit theorems for discretized occupation time functionals," Stochastic Processes and their Applications, Elsevier, vol. 156(C), pages 101-125.

    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. David Baños & Salvador Ortiz-Latorre & Andrey Pilipenko & Frank Proske, 2022. "Strong Solutions of Stochastic Differential Equations with Generalized Drift and Multidimensional Fractional Brownian Initial Noise," Journal of Theoretical Probability, Springer, vol. 35(2), pages 714-771, June.
    2. Mishura, Yu. & Nualart, D., 2004. "Weak solutions for stochastic differential equations with additive fractional noise," Statistics & Probability Letters, Elsevier, vol. 70(4), pages 253-261, December.
    3. Kęstutis Kubilius, 2024. "The Implicit Euler Scheme for FSDEs with Stochastic Forcing: Existence and Uniqueness of the Solution," Mathematics, MDPI, vol. 12(16), pages 1-18, August.
    4. Kubilius, K. & Skorniakov, V., 2016. "On some estimators of the Hurst index of the solution of SDE driven by a fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 109(C), pages 159-167.
    5. Gassiat, Paul & Mądry, Łukasz, 2023. "Perturbations of singular fractional SDEs," Stochastic Processes and their Applications, Elsevier, vol. 161(C), pages 137-172.
    6. Boufoussi, Brahim & Hajji, Salah, 2012. "Neutral stochastic functional differential equations driven by a fractional Brownian motion in a Hilbert space," Statistics & Probability Letters, Elsevier, vol. 82(8), pages 1549-1558.
    7. Marc Mukendi Mpanda, 2022. "Malliavin differentiability of fractional Heston-type model and applications to option pricing," Papers 2207.10709, arXiv.org, revised Aug 2022.
    8. Fan, Xiliang & Yu, Ting & Yuan, Chenggui, 2023. "Asymptotic behaviors for distribution dependent SDEs driven by fractional Brownian motions," Stochastic Processes and their Applications, Elsevier, vol. 164(C), pages 383-415.
    9. Xu, Liping & Luo, Jiaowan, 2018. "Stochastic differential equations driven by fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 142(C), pages 102-108.
    10. León, Jorge A. & Villa, José, 2011. "An Osgood criterion for integral equations with applications to stochastic differential equations with an additive noise," Statistics & Probability Letters, Elsevier, vol. 81(4), pages 470-477, April.
    11. Azmoodeh Ehsan & Mishura Yuliya & Valkeila Esko, 2009. "On hedging European options in geometric fractional Brownian motion market model," Statistics & Risk Modeling, De Gruyter, vol. 27(02), pages 129-144, December.
    12. Sun, Xiaoxia & Guo, Feng, 2015. "On integration by parts formula and characterization of fractional Ornstein–Uhlenbeck process," Statistics & Probability Letters, Elsevier, vol. 107(C), pages 170-177.
    13. Li, Zhi & Yan, Litan, 2018. "Harnack inequalities for SDEs driven by subordinator fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 134(C), pages 45-53.
    14. Coffie, Emmanuel & Duedahl, Sindre & Proske, Frank, 2023. "Sensitivity analysis with respect to a stochastic stock price model with rough volatility via a Bismut–Elworthy–Li formula for singular SDEs," Stochastic Processes and their Applications, Elsevier, vol. 156(C), pages 156-195.
    15. Marc Mukendi Mpanda & Safari Mukeru & Mmboniseni Mulaudzi, 2020. "Generalisation of Fractional-Cox-Ingersoll-Ross Process," Papers 2008.07798, arXiv.org, revised Jul 2022.
    16. Pilipenko, Andrey & Proske, Frank Norbert, 2018. "On perturbations of an ODE with non-Lipschitz coefficients by a small self-similar noise," Statistics & Probability Letters, Elsevier, vol. 132(C), pages 62-73.
    17. 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.
    18. Jan Gairing & Peter Imkeller & Radomyra Shevchenko & Ciprian Tudor, 2020. "Hurst Index Estimation in Stochastic Differential Equations Driven by Fractional Brownian Motion," Journal of Theoretical Probability, Springer, vol. 33(3), pages 1691-1714, September.
    19. Kohei Chiba, 2019. "Estimation of the lead–lag parameter between two stochastic processes driven by fractional Brownian motions," Statistical Inference for Stochastic Processes, Springer, vol. 22(3), pages 323-357, October.
    20. Nourdin, Ivan & Peccati, Giovanni & Viens, Frederi G., 2014. "Comparison inequalities on Wiener space," Stochastic Processes and their Applications, Elsevier, vol. 124(4), pages 1566-1581.

    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:126:y:2016:i:8:p:2323-2366. 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.