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Itô's formula with respect to fractional Brownian motion and its application

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  • W. Dai
  • C. C. Heyde

Abstract

Fractional Brownian motion (FBM) with Hurst index 1 / 2 < H < 1 is not a semimartingale. Consequently, the standard Itô calculus is not available for stochastic integrals with respect to FBM as an integrator if 1 / 2 < H < 1 . In this paper we derive a version of Itô's formula for fractional Brownian motion. Then, as an application, we propose and study a fractional Brownian Scholes stochastic model which includes the standard Black-Scholes model as a special case and is able to account for long range dependence in modeling the price of a risky asset. This article is dedicated to the memory of Roland L. Dobrushin.

Suggested Citation

  • W. Dai & C. C. Heyde, 1996. "Itô's formula with respect to fractional Brownian motion and its application," International Journal of Stochastic Analysis, Hindawi, vol. 9, pages 1-10, January.
    • Handle: RePEc:hin:jnijsa:541390
    DOI: 10.1155/S104895339600038X
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    Cited by:

    1. Manabu Asai & Michael McAleer, 2017. "A fractionally integrated Wishart stochastic volatility model," Econometric Reviews, Taylor & Francis Journals, vol. 36(1-3), pages 42-59, March.
    2. David Nualart & Youssef Ouknine, 2003. "Besov Regularity of Stochastic Integrals with Respect to the Fractional Brownian Motion with Parameter H > 1/2," Journal of Theoretical Probability, Springer, vol. 16(2), pages 451-470, April.
    3. Axel A. Araneda, 2019. "The fractional and mixed-fractional CEV model," Papers 1903.05747, arXiv.org, revised Jun 2019.

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