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On inference for fractional differential equations

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  • Alexandra Chronopoulou
  • Samy Tindel

Abstract

Based on Malliavin calculus tools and approximation results, we show how to compute a maximum likelihood type estimator for a rather general differential equation driven by a fractional Brownian motion with Hurst parameter $$H>1/2$$ . Rates of convergence for the approximation task are provided, and numerical experiments show that our procedure leads to good results in terms of estimation. Copyright Springer Science+Business Media Dordrecht 2013

Suggested Citation

  • Alexandra Chronopoulou & Samy Tindel, 2013. "On inference for fractional differential equations," Statistical Inference for Stochastic Processes, Springer, vol. 16(1), pages 29-61, April.
    • Handle: RePEc:spr:sistpr:v:16:y:2013:i:1:p:29-61
    DOI: 10.1007/s11203-013-9076-z
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    References listed on IDEAS

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    1. 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.
    2. Durham, Garland B & Gallant, A Ronald, 2002. "Numerical Techniques for Maximum Likelihood Estimation of Continuous-Time Diffusion Processes: Reply," Journal of Business & Economic Statistics, American Statistical Association, vol. 20(3), pages 335-338, July.
    3. Baudoin, Fabrice & Coutin, Laure, 2007. "Operators associated with a stochastic differential equation driven by fractional Brownian motions," Stochastic Processes and their Applications, Elsevier, vol. 117(5), pages 550-574, May.
    4. L. C. G. Rogers, 1997. "Arbitrage with Fractional Brownian Motion," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 95-105, January.
    5. Paolo Guasoni, 2006. "No Arbitrage Under Transaction Costs, With Fractional Brownian Motion And Beyond," Mathematical Finance, Wiley Blackwell, vol. 16(3), pages 569-582, July.
    6. Durham, Garland B & Gallant, A Ronald, 2002. "Numerical Techniques for Maximum Likelihood Estimation of Continuous-Time Diffusion Processes," Journal of Business & Economic Statistics, American Statistical Association, vol. 20(3), pages 297-316, July.
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    Citations

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    Cited by:

    1. 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.
    2. Fabienne Comte & Nicolas Marie, 2019. "Nonparametric estimation in fractional SDE," Statistical Inference for Stochastic Processes, Springer, vol. 22(3), pages 359-382, October.
    3. Marie, Nicolas, 2020. "Nonparametric estimation of the trend in reflected fractional SDE," Statistics & Probability Letters, Elsevier, vol. 158(C).
    4. 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.
    5. Fabienne Comte & Nicolas Marie, 2021. "Nonparametric estimation for I.I.D. paths of fractional SDE," Statistical Inference for Stochastic Processes, Springer, vol. 24(3), pages 669-705, October.
    6. Zhang, Pu & Xiao, Wei-lin & Zhang, Xi-li & Niu, Pan-qiang, 2014. "Parameter identification for fractional Ornstein–Uhlenbeck processes based on discrete observation," Economic Modelling, Elsevier, vol. 36(C), pages 198-203.

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