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

  • Ehsan AzmoodehEmail author
  • Esko Valkeila


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.


Fractional Brownian motion Quadratic variation Randomized periodogram 

Mathematics Subject Classification

60G15 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.


  1. Barndorff-Nielsen EO, Shephard N (2002) Estimating quadratic variation using realized variance. J Appl Econ 17:457–477CrossRefGoogle Scholar
  2. Bender C, Sottinen T, Valkeila E (2008) Pricing by hedging and no-arbitrage beyond semimartingales. Finance Stoch 12:441–468MathSciNetzbMATHCrossRefGoogle Scholar
  3. Cheridito P (2001) Mixed fractional Brownian motion. Bernoulli 7(6):913–934MathSciNetzbMATHCrossRefGoogle Scholar
  4. Dasgupta A, Kallianpur G (1999) Multiple fractional integrals. Probab Theory Relat Fields 115:505–525MathSciNetzbMATHCrossRefGoogle Scholar
  5. Dzhaparidze K, Spreij P (1994) Spectral characterization of the optional quadratic variation processes. Stoch Procces Appl 54:165–174MathSciNetzbMATHCrossRefGoogle Scholar
  6. Föllmer H (1981) Calcul d’Ito sans probabilités. Seminar on probability, XV, 143–150, Lect Notes Math, 850, Springer, BerlinGoogle Scholar
  7. Janson S (1997) Gaussian Hilbert spaces. Cambridge University Press, CambridgezbMATHCrossRefGoogle Scholar
  8. Memin J, Mishura Y, Valkeila E (2001) Inequalities for the moments of Wiener integrals with respect to a fractional Brownian motion. Stat Probab Lett 51:197–206MathSciNetzbMATHCrossRefGoogle Scholar
  9. Mishura Y (2008) Stochastic calculus for fractional Brownian Motion and related processes, Lect Notes Math, vol. 1929, Springer, BerlinGoogle Scholar
  10. Nualart D, Răs̨canu A (2002) Differential equations driven by fractional Brownian motion. Collect Math 53:55–81MathSciNetzbMATHGoogle Scholar
  11. Nualart D (2005) The Malliavin calculus and related topics. Springer, BerlinGoogle Scholar
  12. Protter P (2003) Stochastic integration and differential equations. Springer, BerlinGoogle Scholar
  13. Samko SG, Kilbas AA, Marichev OI (1993) Fractional integrals and derivatives, theory and applications. Gordon and Breach Science Publishers, YvendonzbMATHGoogle Scholar
  14. Sondermann D (2006) Introduction to stochastic calculus for finance, A new didactic approach, Lect Note Econ Math Syst, 579, Springer, BerlinGoogle Scholar
  15. Young LC (1936) An inequality of the Hölder type, connected with Stieltjes integration. Acta Math 67:251–282MathSciNetCrossRefGoogle Scholar
  16. Zähle M (1998) Integration with respect to fractal functions and stochastic calculus. I. Probab Theory Relat Fields 111:333–372zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Department of Mathematics and Systems AnalysisAalto University AaltoFinland

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