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Full characterization of the fractional Poisson process


  • Mauro Politi
  • Taisei Kaizoji
  • Enrico Scalas


The fractional Poisson process (FPP) is a counting process with independent and identically distributed inter-event times following the Mittag-Leffler distribution. This process is very useful in several fields of applied and theoretical physics including models for anomalous diffusion. Contrary to the well-known Poisson process, the fractional Poisson process does not have stationary and independent increments. It is not a L\'evy process and it is not a Markov process. In this letter, we present formulae for its finite-dimensional distribution functions, fully characterizing the process. These exact analytical results are compared to Monte Carlo simulations.

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  • Mauro Politi & Taisei Kaizoji & Enrico Scalas, 2011. "Full characterization of the fractional Poisson process," Papers 1104.4234,
  • Handle: RePEc:arx:papers:1104.4234

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    References listed on IDEAS

    1. Haydyn Brown & David Hobson & L. C. G. Rogers, 2001. "Robust Hedging of Barrier Options," Mathematical Finance, Wiley Blackwell, vol. 11(3), pages 285-314.
    2. Ian Martin, 2011. "Simple Variance Swaps," NBER Working Papers 16884, National Bureau of Economic Research, Inc.
    3. Robert Jarrow & Younes Kchia & Martin Larsson & Philip Protter, 2013. "Discretely sampled variance and volatility swaps versus their continuous approximations," Finance and Stochastics, Springer, vol. 17(2), pages 305-324, April.
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    Cited by:

    1. De Martino, Giuseppe & Spina, Serena, 2015. "Exploiting the time-dynamics of news diffusion on the Internet through a generalized Susceptible–Infected model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 438(C), pages 634-644.
    2. repec:eee:stapro:v:129:y:2017:i:c:p:260-268 is not listed on IDEAS
    3. Beghin, Luisa & Macci, Claudio, 2013. "Large deviations for fractional Poisson processes," Statistics & Probability Letters, Elsevier, vol. 83(4), pages 1193-1202.
    4. Leonenko, Nikolai & Scalas, Enrico & Trinh, Mailan, 2017. "The fractional non-homogeneous Poisson process," Statistics & Probability Letters, Elsevier, vol. 120(C), pages 147-156.
    5. Orsingher, Enzo & Polito, Federico, 2013. "On the integral of fractional Poisson processes," Statistics & Probability Letters, Elsevier, vol. 83(4), pages 1006-1017.
    6. Chicheportiche, Rémy & Chakraborti, Anirban, 2017. "A model-free characterization of recurrences in stationary time series," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 474(C), pages 312-318.
    7. Orsingher, Enzo & Polito, Federico, 2012. "The space-fractional Poisson process," Statistics & Probability Letters, Elsevier, vol. 82(4), pages 852-858.

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