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Fractional Randomness and the Brownian Bridge

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  • Tapiero, Charles S.
  • Vallois, Pierre

Abstract

This paper introduces a statistical approach to fractional randomness based on the Central Limit Theorem. We show under general conditions that fractional noise-randomness defined relative to a uniform distribution, implies as well a fractional Brownian Bridge randomness rather than a Fractional Brownian Motion. We analyze further their fractional properties, namely, their variance and covariance and obtain specific results for particular distributions including the fractional uniform distribution and an exponential distribution. The results we obtain have both practical and theoretical implications to the many applications of fractional calculus and in particular, when they are applied to modeling statistical problems where time scaling and randomness prime. This is the case in finance, insurance and risk models as well as in other areas of interest.

Suggested Citation

  • Tapiero, Charles S. & Vallois, Pierre, 2018. "Fractional Randomness and the Brownian Bridge," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 503(C), pages 835-843.
    • Handle: RePEc:eee:phsmap:v:503:y:2018:i:c:p:835-843
    DOI: 10.1016/j.physa.2018.02.097
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    References listed on IDEAS

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    1. Robert J. Elliott & John Van Der Hoek, 2003. "A General Fractional White Noise Theory And Applications To Finance," Mathematical Finance, Wiley Blackwell, vol. 13(2), pages 301-330, April.
    2. Tapiero, Charles S. & Vallois, Pierre, 2016. "Fractional randomness," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 462(C), pages 1161-1177.
    3. Tomas Björk & Henrik Hult, 2005. "A note on Wick products and the fractional Black-Scholes model," Finance and Stochastics, Springer, vol. 9(2), pages 197-209, April.
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    Cited by:

    1. Tapiero, Charles S. & Vallois, Pierre, 2018. "Randomness and fractional stable distributions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 511(C), pages 54-60.

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