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Fractional diffusion equation for a power-law-truncated Lévy process

Author

Listed:
  • Sokolov, I.M
  • Chechkin, A.V
  • Klafter, J

Abstract

Truncated Lévy flights are stochastic processes which display a crossover from a heavy-tailed Lévy behavior to a faster decaying probability distribution function (pdf). Putting less weight on long flights overcomes the divergence of the Lévy distribution second moment. We introduce a fractional generalization of the diffusion equation, whose solution defines a process in which a Lévy flight of exponent α is truncated by a power-law of exponent 5−α. A closed form for the characteristic function of the process is derived. The pdf of the displacement slowly converges to a Gaussian in its central part showing however a power-law far tail. Possible applications are discussed.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:phsmap:v:336:y:2004:i:3:p:245-251
    DOI: 10.1016/j.physa.2003.12.044
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    Citations

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    Cited by:

    1. Sebastian, Orzeł & Agnieszka, Wyłomańska, 2010. "Calibration of the subdiffusive arithmetic Brownian motion with tempered stable waiting-times," MPRA Paper 28593, University Library of Munich, Germany.
    2. Cartea, Álvaro & del-Castillo-Negrete, Diego, 2007. "Fractional diffusion models of option prices in markets with jumps," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 374(2), pages 749-763.
    3. Sandev, Trifce & Sokolov, Igor M. & Metzler, Ralf & Chechkin, Aleksei, 2017. "Beyond monofractional kinetics," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 210-217.
    4. Romanovsky, M.Yu. & Vidov, P.V., 2011. "Analytical representation of stock and stock-indexes returns: Non-Gaussian random walks with various jump laws," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(21), pages 3794-3805.
    5. Chakrabarty, Arijit & Meerschaert, Mark M., 2011. "Tempered stable laws as random walk limits," Statistics & Probability Letters, Elsevier, vol. 81(8), pages 989-997, August.
    6. Zhang, Yuxin & Li, Qian & Ding, Hengfei, 2018. "High-order numerical approximation formulas for Riemann-Liouville (Riesz) tempered fractional derivatives: construction and application (I)," Applied Mathematics and Computation, Elsevier, vol. 329(C), pages 432-443.
    7. Garanina, O.S. & Romanovsky, M.Yu., 2015. "New multi-parametric analytical approximations of exponential distribution with power law tails for new cars sells and other applications," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 427(C), pages 1-9.
    8. Meerschaert, Mark M. & Scheffler, Hans-Peter, 2006. "Stochastic model for ultraslow diffusion," Stochastic Processes and their Applications, Elsevier, vol. 116(9), pages 1215-1235, September.
    9. Maike A. F. dos Santos, 2019. "Mittag–Leffler Memory Kernel in Lévy Flights," Mathematics, MDPI, vol. 7(9), pages 1-13, August.

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