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Tempered stable laws as random walk limits

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  • Chakrabarty, Arijit
  • Meerschaert, Mark M.

Abstract

Stable laws can be tempered by modifying the Lévy measure to cool the probability of large jumps. Tempered stable laws retain their signature power law behavior at infinity, and infinite divisibility. This paper develops random walk models that converge to a tempered stable law under a triangular array scheme. Since tempered stable laws and processes are useful in statistical physics, these random walk models can provide a basic physical model for the underlying physical phenomena.

Suggested Citation

  • Chakrabarty, Arijit & Meerschaert, Mark M., 2011. "Tempered stable laws as random walk limits," Statistics & Probability Letters, Elsevier, vol. 81(8), pages 989-997, August.
    • Handle: RePEc:eee:stapro:v:81:y:2011:i:8:p:989-997
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    References listed on IDEAS

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    1. Aban, Inmaculada B. & Meerschaert, Mark M. & Panorska, Anna K., 2006. "Parameter Estimation for the Truncated Pareto Distribution," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 270-277, March.
    2. Sokolov, I.M & Chechkin, A.V & Klafter, J, 2004. "Fractional diffusion equation for a power-law-truncated Lévy process," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 336(3), pages 245-251.
    3. Peter Carr & Hélyette Geman & Dilip B. Madan & Marc Yor, 2003. "Stochastic Volatility for Lévy Processes," Mathematical Finance, Wiley Blackwell, vol. 13(3), pages 345-382, July.
    4. Peter Carr & Helyette Geman, 2002. "The Fine Structure of Asset Returns: An Empirical Investigation," The Journal of Business, University of Chicago Press, vol. 75(2), pages 305-332, April.
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    Cited by:

    1. Hernández-Hernández, M.E. & Kolokoltsov, V.N. & Toniazzi, L., 2017. "Generalised fractional evolution equations of Caputo type," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 184-196.
    2. Antoine Jacquier & Lorenzo Torricelli, 2019. "Anomalous diffusions in option prices: connecting trade duration and the volatility term structure," Papers 1908.03007, arXiv.org, revised Apr 2020.
    3. Du, Qiang & Toniazzi, Lorenzo & Zhou, Zhi, 2020. "Stochastic representation of solution to nonlocal-in-time diffusion," Stochastic Processes and their Applications, Elsevier, vol. 130(4), pages 2058-2085.
    4. Meerschaert, Mark M. & Toaldo, Bruno, 2019. "Relaxation patterns and semi-Markov dynamics," Stochastic Processes and their Applications, Elsevier, vol. 129(8), pages 2850-2879.

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