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Estimation of the volatility persistence in a discretely observed diffusion model

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  • Rosenbaum, Mathieu

Abstract

We consider the stochastic volatility model with B a Brownian motion and [sigma] of the form where WH is a fractional Brownian motion, independent of the driving Brownian motion B, with Hurst parameter H>=1/2. This model allows for persistence in the volatility [sigma]. The parameter of interest is H. The functions [Phi], a and f are treated as nuisance parameters and [xi]0 is a random initial condition. For a fixed objective time T, we construct from discrete data Yi/n,i=0,...,nT, a wavelet based estimator of H, inspired by adaptive estimation of quadratic functionals. We show that the accuracy of our estimator is n-1/(4H+2) and that this rate is optimal in a minimax sense.

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  • Rosenbaum, Mathieu, 2008. "Estimation of the volatility persistence in a discretely observed diffusion model," Stochastic Processes and their Applications, Elsevier, vol. 118(8), pages 1434-1462, August.
    • Handle: RePEc:eee:spapps:v:118:y:2008:i:8:p:1434-1462
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    Cited by:

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    3. Nourdin, Ivan & Diu Tran, T.T., 2019. "Statistical inference for Vasicek-type model driven by Hermite processes," Stochastic Processes and their Applications, Elsevier, vol. 129(10), pages 3774-3791.
    4. Pavel Kříž & Leszek Szała, 2020. "The Combined Estimator for Stochastic Equations on Graphs with Fractional Noise," Mathematics, MDPI, vol. 8(10), pages 1-21, October.
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    6. Masaaki Fukasawa & Tetsuya Takabatake & Rebecca Westphal, 2019. "Is Volatility Rough ?," Papers 1905.04852, arXiv.org, revised May 2019.
    7. Archil Gulisashvili & Frederi Viens & Xin Zhang, 2015. "Small-time asymptotics for Gaussian self-similar stochastic volatility models," Papers 1505.05256, arXiv.org, revised Mar 2016.
    8. Bianchi, Sergio & Pianese, Augusto, 2018. "Time-varying Hurst–Hölder exponents and the dynamics of (in)efficiency in stock markets," Chaos, Solitons & Fractals, Elsevier, vol. 109(C), pages 64-75.
    9. Masaaki Fukasawa & Tetsuya Takabatake & Rebecca Westphal, 2022. "Consistent estimation for fractional stochastic volatility model under high‐frequency asymptotics," Mathematical Finance, Wiley Blackwell, vol. 32(4), pages 1086-1132, October.
    10. Sixian Jin & Qidi Peng & Henry Schellhorn, 2018. "Estimation of the pointwise Hölder exponent of hidden multifractional Brownian motion using wavelet coefficients," Statistical Inference for Stochastic Processes, Springer, vol. 21(1), pages 113-140, April.
    11. Peng, Qidi & Zhao, Ran, 2018. "A general class of multifractional processes and stock price informativeness," Chaos, Solitons & Fractals, Elsevier, vol. 115(C), pages 248-267.
    12. Pavel Kříž & Leszek Szała, 2020. "Least-Squares Estimators of Drift Parameter for Discretely Observed Fractional Ornstein–Uhlenbeck Processes," Mathematics, MDPI, vol. 8(5), pages 1-20, May.

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