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Prabhakar Lévy processes

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  • Gajda, Janusz
  • Beghin, Luisa

Abstract

We introduce here a generalization of the Mittag-Leffler Lévy process (with parameter α), obtained by extending its Lévy measure through the Prabhakar function (which is a Mittag-Leffler with the additional parameters β and γ). We prove that this so-called Prabhakar process, in the special case β=1, can be represented as an α-stable process subordinated by an independent generalized gamma subordinator; thus it can be considered as an extension of the geometric stable process, to which it reduces for γ=1. On the other hand, for α=β=1, it coincides with the generalized gamma process itself. Therefore, by suitably specifying the three parameters, the Prabhakar process turns out to represent an interpolation among various well-known and widely applied stochastic models.

Suggested Citation

  • Gajda, Janusz & Beghin, Luisa, 2021. "Prabhakar Lévy processes," Statistics & Probability Letters, Elsevier, vol. 178(C).
    • Handle: RePEc:eee:stapro:v:178:y:2021:i:c:s0167715221001243
    DOI: 10.1016/j.spl.2021.109162
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    References listed on IDEAS

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    1. Beghin, L., 2012. "Random-time processes governed by differential equations of fractional distributed order," Chaos, Solitons & Fractals, Elsevier, vol. 45(11), pages 1314-1327.
    2. Madan, Dilip B & Seneta, Eugene, 1990. "The Variance Gamma (V.G.) Model for Share Market Returns," The Journal of Business, University of Chicago Press, vol. 63(4), pages 511-524, October.
    3. Neil Shephard & Ole E. Barndorff-Nielsen, 2012. "Basics of Levy processes," Economics Series Working Papers 610, University of Oxford, Department of Economics.
    4. Beghin, Luisa, 2018. "Fractional diffusion-type equations with exponential and logarithmic differential operators," Stochastic Processes and their Applications, Elsevier, vol. 128(7), pages 2427-2447.
    5. H. J. Haubold & A. M. Mathai & R. K. Saxena, 2011. "Mittag-Leffler Functions and Their Applications," Journal of Applied Mathematics, Hindawi, vol. 2011, pages 1-51, May.
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