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Generalized Counting Processes in a Stochastic Environment

Author

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  • Davide Cocco

    (Dipartimento SBAI, La Sapienza Università di Roma, Via Antonio Scarpa 16, 00161 Roma, Italy)

  • Massimiliano Giona

    (DICMA, La Sapienza Università di Roma, Via Eudossiana 18, 00184 Roma, Italy)

Abstract

This paper addresses the generalization of counting processes through the age formalism of Lévy Walks. Simple counting processes are introduced and their properties are analyzed: Poisson processes or fractional Poisson processes can be recovered as particular cases. The stationarity assumption in the renewal mechanism characterizing simple counting processes can be modified in different ways, leading to the definition of generalized counting processes. In the case that the transition mechanism of a counting process depends on the environmental conditions—i.e., the parameters describing the occurrence of new events are themselves stochastic processes—the counting processes is said to be influenced by environmental stochasticity. The properties of this class of processes are analyzed, providing several examples and applications and showing the occurrence of new phenomena related to the modulation of the long-term scaling exponent by environmental noise.

Suggested Citation

  • Davide Cocco & Massimiliano Giona, 2021. "Generalized Counting Processes in a Stochastic Environment," Mathematics, MDPI, vol. 9(20), pages 1-19, October.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:20:p:2573-:d:655704
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    References listed on IDEAS

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    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. Stirzaker, David, 2005. "Stochastic Processes and Models," OUP Catalogue, Oxford University Press, number 9780198568148.
    6. Mauro Politi & Taisei Kaizoji & Enrico Scalas, 2011. "Full characterization of the fractional Poisson process," Papers 1104.4234, arXiv.org.
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