Spectral characterization of the quadratic variation of mixed Brownian–fractional Brownian motion
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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.
KeywordsFractional Brownian motion Quadratic variation Randomized periodogram
Mathematics Subject Classification60G15 62M15
Azmoodeh is grateful to Finnish Graduate School in Stochastic and Statistic (FGSS) for financial support, and Valkeila acknowledges the support of Academy of Finland, Grant 212875. We are grateful to an anonymous referee for a careful reading of the two versions of this paper.
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