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Spectral characterization of the quadratic variation of mixed Brownian–fractional Brownian motion

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  • Ehsan Azmoodeh
  • Esko Valkeila

Abstract

Dzhaparidze and Spreij (Stoch Process Appl, 54:165–174, 1994 ) showed that the quadratic variation of a semimartingale can be approximated using a randomized periodogram. We show that the same approximation is valid for a special class of continuous stochastic processes. This class contains both semimartingales and non-semimartingales. The motivation comes partially from the recent work by Bender et al. (Finance Stoch, 12:441–468, 2008 ), where it is shown that the quadratic variation of the log-returns determines the hedging strategy. Copyright Springer Science+Business Media Dordrecht 2013

Suggested Citation

  • Ehsan Azmoodeh & Esko Valkeila, 2013. "Spectral characterization of the quadratic variation of mixed Brownian–fractional Brownian motion," Statistical Inference for Stochastic Processes, Springer, vol. 16(2), pages 97-112, July.
    • Handle: RePEc:spr:sistpr:v:16:y:2013:i:2:p:97-112
    DOI: 10.1007/s11203-013-9079-9
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    References listed on IDEAS

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    1. Mémin, Jean & Mishura, Yulia & Valkeila, Esko, 2001. "Inequalities for the moments of Wiener integrals with respect to a fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 51(2), pages 197-206, January.
    2. Christian Bender & Tommi Sottinen & Esko Valkeila, 2008. "Pricing by hedging and no-arbitrage beyond semimartingales," Finance and Stochastics, Springer, vol. 12(4), pages 441-468, October.
    3. Ole E. Barndorff-Nielsen & Neil Shephard, 2002. "Estimating quadratic variation using realized variance," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 17(5), pages 457-477.
    4. Dzhaparidze, Kacha & Spreij, Peter, 1994. "Spectral characterization of the optional quadratic variation process," Stochastic Processes and their Applications, Elsevier, vol. 54(1), pages 165-174, November.
    5. Dieter Sondermann, 2006. "Introduction to Stochastic Calculus for Finance," Lecture Notes in Economics and Mathematical Systems, Springer, number 978-3-540-34837-5, October.
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