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Regularization of differential equations by fractional noise

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  • Nualart, David
  • Ouknine, Youssef

Abstract

Let {BtH,t[set membership, variant][0,T]} be a fractional Brownian motion with Hurst parameter H. We prove the existence and uniqueness of a strong solution for a stochastic differential equation of the form , where b(s,x) is a bounded Borel function with linear growth in x (case ) or a Hölder continuous function of order strictly larger than 1-1/2H in x and than in time (case ).

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:102:y:2002:i:1:p:103-116
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    Cited by:

    1. 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.
    2. 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.
    3. 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.
    4. 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.
    5. 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.
    6. Liu, Yanghui & Nualart, Eulalia & Tindel, Samy, 2019. "LAN property for stochastic differential equations with additive fractional noise and continuous time observation," Stochastic Processes and their Applications, Elsevier, vol. 129(8), pages 2880-2902.
    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. 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.
    9. 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.
    10. 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.
    11. 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.
    12. 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.
    13. Hu, Yaozhong & Nualart, David & Song, Xiaoming, 2008. "A singular stochastic differential equation driven by fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 78(14), pages 2075-2085, October.
    14. 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.
    15. 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.
    16. 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.
    17. Marc Mukendi Mpanda & Safari Mukeru & Mmboniseni Mulaudzi, 2020. "Generalisation of Fractional-Cox-Ingersoll-Ross Process," Papers 2008.07798, arXiv.org, revised Jul 2022.
    18. 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.
    19. Es-Sebaiy, Khalifa & Ouassou, Idir & Ouknine, Youssef, 2009. "Estimation of the drift of fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 79(14), pages 1647-1653, July.
    20. Xu, Liping & Luo, Jiaowan, 2018. "Stochastic differential equations driven by fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 142(C), pages 102-108.
    21. 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.
    22. Gassiat, Paul & Mądry, Łukasz, 2023. "Perturbations of singular fractional SDEs," Stochastic Processes and their Applications, Elsevier, vol. 161(C), pages 137-172.
    23. 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.
    24. 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.
    25. 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.

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