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Fractional normal inverse Gaussian diffusion

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  • Kumar, A.
  • Meerschaert, Mark M.
  • Vellaisamy, P.

Abstract

A fractional normal inverse Gaussian (FNIG) process is a fractional Brownian motion subordinated to an inverse Gaussian process. This paper shows how the FNIG process emerges naturally as the limit of a random walk with correlated jumps separated by i.i.d. waiting times. Similarly, we show that the NIG process, a Brownian motion subordinated to an inverse Gaussian process, is the limit of a random walk with uncorrelated jumps separated by i.i.d. waiting times. The FNIG process is also derived as the limit of a fractional ARIMA processes. Finally, the NIG densities are shown to solve the relativistic diffusion equation from statistical physics.

Suggested Citation

  • Kumar, A. & Meerschaert, Mark M. & Vellaisamy, P., 2011. "Fractional normal inverse Gaussian diffusion," Statistics & Probability Letters, Elsevier, vol. 81(1), pages 146-152, January.
    • Handle: RePEc:eee:stapro:v:81:y:2011:i:1:p:146-152
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    References listed on IDEAS

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    1. Meerschaert, Mark M. & Nane, Erkan & Xiao, Yimin, 2009. "Correlated continuous time random walks," Statistics & Probability Letters, Elsevier, vol. 79(9), pages 1194-1202, May.
    2. Enrico Scalas, 2006. "Five Years of Continuous-time Random Walks in Econophysics," Lecture Notes in Economics and Mathematical Systems, in: Akira Namatame & Taisei Kaizouji & Yuuji Aruka (ed.), The Complex Networks of Economic Interactions, pages 3-16, Springer.
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    Cited by:

    1. Sung Ik Kim, 2022. "ARMA–GARCH model with fractional generalized hyperbolic innovations," Financial Innovation, Springer;Southwestern University of Finance and Economics, vol. 8(1), pages 1-25, December.
    2. Gajda, J. & Kumar, A. & Wyłomańska, A., 2019. "Stable Lévy process delayed by tempered stable subordinator," Statistics & Probability Letters, Elsevier, vol. 145(C), pages 284-292.
    3. Kumar, A. & Nane, Erkan & Vellaisamy, P., 2011. "Time-changed Poisson processes," Statistics & Probability Letters, Elsevier, vol. 81(12), pages 1899-1910.

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