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Representations of fractional Brownian motion using vibrating strings

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  • Dzhaparidze, Kacha
  • van Zanten, Harry
  • Zareba, Pawel

Abstract

In this paper, we show that the moving average and series representations of fractional Brownian motion can be obtained using the spectral theory of vibrating strings. The representations are shown to be consequences of general theorems valid for a large class of second-order processes with stationary increments. Specifically, we use the 1-1 relation discovered by M.G. Krein between spectral measures of continuous second-order processes with stationary increments and differential equations describing the vibrations of a string with a certain length and mass distribution.

Suggested Citation

  • Dzhaparidze, Kacha & van Zanten, Harry & Zareba, Pawel, 2005. "Representations of fractional Brownian motion using vibrating strings," Stochastic Processes and their Applications, Elsevier, vol. 115(12), pages 1928-1953, December.
    • Handle: RePEc:eee:spapps:v:115:y:2005:i:12:p:1928-1953
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    References listed on IDEAS

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    1. Novikov, Alexander & Valkeila, Esko, 1999. "On some maximal inequalities for fractional Brownian motions," Statistics & Probability Letters, Elsevier, vol. 44(1), pages 47-54, August.
    2. Baudoin, Fabrice & Nualart, David, 2003. "Equivalence of Volterra processes," Stochastic Processes and their Applications, Elsevier, vol. 107(2), pages 327-350, October.
    3. Hult, Henrik, 2003. "Approximating some Volterra type stochastic integrals with applications to parameter estimation," Stochastic Processes and their Applications, Elsevier, vol. 105(1), pages 1-32, May.
    4. 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.
    5. Dzhaparidze, K. & Ferreira, J. A., 2002. "A frequency domain approach to some results on fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 60(2), pages 155-168, November.
    6. Cheridito, Patrick, 2004. "Gaussian moving averages, semimartingales and option pricing," Stochastic Processes and their Applications, Elsevier, vol. 109(1), pages 47-68, January.
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    1. van Zanten, Harry, 2007. "When is a linear combination of independent fBm's equivalent to a single fBm?," Stochastic Processes and their Applications, Elsevier, vol. 117(1), pages 57-70, January.

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