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Stochastic volatility and fractional Brownian motion

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  • Gloter, A.
  • Hoffmann, M.

Abstract

We observe (Yt) at times i/n, i=0,...,n, in the parametric stochastic volatility modeldYt=[Phi]([theta],WtH) dWt,where (Wt) is a Brownian motion, independent of the fractional Brownian motion (WtH) with Hurst parameter . The sample size n increases not because of a longer observation period, but rather, because of more frequent observations. We prove that the unusual rate n-1/(4H+2) is asymptotically optimal for estimating the one-dimensional parameter [theta], and we construct a contrast estimator based on an approximation of a suitably normalized quadratic variation that achieves the optimal rate.

Suggested Citation

  • Gloter, A. & Hoffmann, M., 2004. "Stochastic volatility and fractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 113(1), pages 143-172, September.
    • Handle: RePEc:eee:spapps:v:113:y:2004:i:1:p:143-172
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    Cited by:

    1. 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.
    2. León, José & Ludeña, Carenne, 2007. "Limits for weighted p-variations and likewise functionals of fractional diffusions with drift," Stochastic Processes and their Applications, Elsevier, vol. 117(3), pages 271-296, March.
    3. José León & Carenne Ludeña, 2015. "Difference based estimators and infill statistics," Statistical Inference for Stochastic Processes, Springer, vol. 18(1), pages 1-31, April.
    4. Barndorff-Nielsen, Ole E. & Shephard, Neil, 2006. "Impact of jumps on returns and realised variances: econometric analysis of time-deformed Levy processes," Journal of Econometrics, Elsevier, vol. 131(1-2), pages 217-252.

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