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Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion

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  • Nualart, David
  • Saussereau, Bruno

Abstract

We prove the Malliavin regularity of the solution of a stochastic differential equation driven by a fractional Brownian motion of Hurst parameter H>0.5. The result is based on the Fréchet differentiability with respect to the input function for deterministic differential equations driven by Hölder continuous functions. It is also shown that the law of the solution has a density with respect to the Lebesgue measure, under a suitable nondegeneracy condition.

Suggested Citation

  • Nualart, David & Saussereau, Bruno, 2009. "Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 119(2), pages 391-409, February.
    • Handle: RePEc:eee:spapps:v:119:y:2009:i:2:p:391-409
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    References listed on IDEAS

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    1. Nourdin, Ivan & Simon, Thomas, 2006. "On the absolute continuity of one-dimensional SDEs driven by a fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 76(9), pages 907-912, May.
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    Cited by:

    1. Bondarenko, Valeria & Bondarenko, Victor & Truskovskyi, Kyryl, 2017. "Forecasting of time data with using fractional Brownian motion," Chaos, Solitons & Fractals, Elsevier, vol. 97(C), pages 44-50.
    2. 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.
    3. Armstrong, John & Ionescu, Andrei, 2024. "Itô stochastic differentials," Stochastic Processes and their Applications, Elsevier, vol. 171(C).
    4. Yaozhong Hu & Samy Tindel, 2013. "Smooth Density for Some Nilpotent Rough Differential Equations," Journal of Theoretical Probability, Springer, vol. 26(3), pages 722-749, September.
    5. 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.
    6. Sin, Myong-Guk & Ri, Kyong-Il & Kim, Kyong-Hui, 2022. "Existence and uniqueness of solution for coupled fractional mean-field forward–backward stochastic differential equations," Statistics & Probability Letters, Elsevier, vol. 190(C).
    7. 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.
    8. 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.
    9. Jorge A. León & Samy Tindel, 2012. "Malliavin Calculus for Fractional Delay Equations," Journal of Theoretical Probability, Springer, vol. 25(3), pages 854-889, September.
    10. 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.
    11. 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.
    12. Alexandra Chronopoulou & Samy Tindel, 2013. "On inference for fractional differential equations," Statistical Inference for Stochastic Processes, Springer, vol. 16(1), pages 29-61, April.
    13. Peter Kloeden & Andreas Neuenkirch & Raffaella Pavani, 2011. "Multilevel Monte Carlo for stochastic differential equations with additive fractional noise," Annals of Operations Research, Springer, vol. 189(1), pages 255-276, September.
    14. 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.
    15. Quer-Sardanyons, Lluís & Tindel, Samy, 2012. "Pathwise definition of second-order SDEs," Stochastic Processes and their Applications, Elsevier, vol. 122(2), pages 466-497.
    16. Fan, XiLiang, 2015. "Logarithmic Sobolev inequalities for fractional diffusion," Statistics & Probability Letters, Elsevier, vol. 106(C), pages 165-172.
    17. Eric Djeutcha & Didier Alain Njamen Njomen & Louis-Aimé Fono, 2019. "Solving Arbitrage Problem on the Financial Market Under the Mixed Fractional Brownian Motion With Hurst Parameter H ∈]1/2,3/4[," Journal of Mathematics Research, Canadian Center of Science and Education, vol. 11(1), pages 76-92, February.
    18. Shevchenko, Georgiy & Shalaiko, Taras, 2013. "Malliavin regularity of solutions to mixed stochastic differential equations," Statistics & Probability Letters, Elsevier, vol. 83(12), pages 2638-2646.

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