Generalized Counting Processes in a Stochastic Environment

Davide Cocco and Massimiliano Giona
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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

Mathematics, 2021, vol. 9, issue 20, 1-19

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.

Keywords: counting processes; Lévy walks; age description; environmental stochasticity (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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