Abstract and Applied Analysis

Alternative Forms of Compound Fractional Poisson Processes

Luisa Beghin and Claudio Macci

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We study here different fractional versions of the compound Poisson process. The fractionality is introduced in the counting process representing the number of jumps as well as in the density of the jumps themselves. The corresponding distributions are obtained explicitly and proved to be solution of fractional equations of order less than one. Only in the final case treated in this paper, where the number of jumps is given by the fractional-difference Poisson process defined in Orsingher and Polito (2012), we have a fractional driving equation, with respect to the time argument, with order greater than one. Moreover, in this case, the compound Poisson process is Markovian and this is also true for the corresponding limiting process. All the processes considered here are proved to be compositions of continuous time random walks with stable processes (or inverse stable subordinators). These subordinating relationships hold, not only in the limit, but also in the finite domain. In some cases the densities satisfy master equations which are the fractional analogues of the well-known Kolmogorov one.

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Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 747503, 30 pages.

First available in Project Euclid: 5 April 2013

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Beghin, Luisa; Macci, Claudio. Alternative Forms of Compound Fractional Poisson Processes. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 747503, 30 pages. doi:10.1155/2012/747503. https://projecteuclid.org/euclid.aaa/1365168348

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  • O. N. Repin and A. I. Saichev, “Fractional Poisson law,” Radiophysics and Quantum Electronics, vol. 43, no. 9, pp. 738–741, 2000.
  • F. Mainardi, R. Gorenflo, and E. Scalas, “A fractional generalization of the Poisson processes,” Vietnam Journal of Mathematics, vol. 32, pp. 53–64, 2004.
  • F. Mainardi, R. Gorenflo, and A. Vivoli, “Beyond the Poisson renewal process: a tutorial survey,” Journal of Computational and Applied Mathematics, vol. 205, no. 2, pp. 725–735, 2007.
  • L. Beghin and E. Orsingher, “Fractional Poisson processes and related planar random motions,” Electronic Journal of Probability, vol. 14, no. 61, pp. 1790–1827, 2009.
  • M. Politi, T. Kaizoji, and E. Scalas, “Full characterization of the fractional Poisson process,” Europhysics Letters, vol. 96, no. 2, Article ID 20004, 6 pages, 2011.
  • M. M. Meerschaert, E. Nane, and P. Vellaisamy, “The fractional Poisson process and the inverse stable subordinator,” Electronic Journal of Probability, vol. 16, no. 59, pp. 1600–1620, 2011.
  • G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Processes, Chapman & Hall, New York, NY, USA, 1994.
  • L. Beghin and C. Macci, “Large deviations for fractional čommentComment on ref. [4?]: Please update the information of this reference, if possible. Poisson processes,” submitted,
  • E. Orsingher and F. Polito, “The space-fractional Poisson process,” Statistics & Probability Letters, vol. 82, no. 4, pp. 852–858, 2012.
  • M. M. Meerschaert and H.-P. Scheffler, “Limit theorems for continuous-time random walks with infinite mean waiting times,” Journal of Applied Probability, vol. 41, no. 3, pp. 623–638, 2004.
  • R. Hilfer and L. Anton, “Fractional master equations and fractal random walks,” Physical Review E, vol. 51, no. 2, pp. R848–R851, 1995.
  • A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, The Netherlands, 2006.
  • M. M. Meerschaert, D. A. Benson, H.-P. Scheffler, and B. Baeumer, “Stochastic solution of space-time fractional diffusion equations,” Physical Review E, vol. 65, no. 4, Article ID 041103, 2002.
  • E. Scalas, “The application of continuous-time random walks in finance and economics,” Physica A, vol. 362, no. 2, pp. 225–239, 2006.
  • E. Scalas, “A class of CTRWs: compound fractional Poisson processes,” in Fractional Dynamics, pp. 353–374, World Scientific Publishers, Hackensack, NJ, USA, 2012.
  • F. Mainardi, Y. Luchko, and G. Pagnini, “The fundamental solution of the space-time fractional diffusion equation,” Fractional Calculus & Applied Analysis, vol. 4, no. 2, pp. 153–192, 2001.
  • B. Baeumer, M. M. Meerschaert, and E. Nane, “Space-time duality for fractional diffusion,” Journal of Applied Probability, vol. 46, no. 4, pp. 1100–1115, 2009.
  • T. Rolski, H. Schmidli, V. Schmidt, and J. Teugels, Stochastic Processes for Insurance and Finance, Wiley Series in Probability and Statistics, John Wiley & Sons, Chichester, UK, 1999.
  • G. K. Smyth and B. Jørgensen, “Fitting Tweedie's compound Poisson model to insurance claims data: dispersion modelling,” Astin Bulletin, vol. 32, no. 1, pp. 143–157, 2002.
  • C. S. Withers and S. Nadarajah, “On the compound Poisson-gamma distribution,” Kybernetika, vol. 47, no. 1, pp. 15–37, 2011.
  • L. Beghin and E. Orsingher, “Poisson-type processes governed by fractional and higher-order recursive differential equations,” Electronic Journal of Probability, vol. 15, no. 22, pp. 684–709, 2010.
  • A. De Gregorio and E. Orsingher, “Flying randomly in ${\mathbb{R}}^{d}$ with Dirichlet displacements,” Stochastic Processes and their Applications, vol. 122, no. 2, pp. 676–713, 2012.
  • N. L. Johnson, S. Kotz, and A. W. Kemp, Univariate Discrete Distributions, Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, John Wiley & Sons, New York, NY, USA, Second edition, 1992.
  • N. Balakrishnan and T. J. Kozubowski, “A class of weighted Poisson processes,” Statistics & Probability Letters, vol. 78, no. 15, pp. 2346–2352, 2008.
  • S. Gerhold, “Asymptotics for a variant of the Mittag-Leffler function,” Integral Transforms and Special Functions, vol. 23, no. 6, pp. 397–403, 2012.
  • H. J. Haubold, A. M. Mathai, and R. K. Saxena, “Mittag-Leffler functions and their applications,” Journal of Applied Mathematics, vol. 2011, Article ID 298628, 51 pages, 2011.
  • K. Jayakumar and R. P. Suresh, “Mittag-Leffler distributions,” Journal of Indian Society of Probability and Statistics, vol. 7, pp. 51–71, 2003.
  • L. Beghin, “Fractional relaxation equations and Brownian crossing probabilities of a random boundary,” Advances in Applied Probability, vol. 44, no. 2, pp. 479–505, 2012.
  • M. D'Ovidio, “From Sturm-Liouville problems to fractional and anomalous diffusions,” Stochastic Processes and their Applications, vol. 122, no. 10, pp. 3513–3544, 2012.
  • M. D'Ovidio, “Explicit solutions to fractional diffusion equations via generalized gamma convolution,” Electronic Communications in Probability, vol. 15, pp. 457–474, 2010.
  • L. F. James, “Lamperti-type laws,” The Annals of Applied Probability, vol. 20, no. 4, pp. 1303–1340, 2010.
  • M. D'Ovidio, “On the fractional counterpart of the higher-order equations,” Statistics & Probability Letters, vol. 81, no. 12, pp. 1929–1939, 2011.