We characterize a Hawkes point process with kernel proportional to the probability density function of Mittag-Leffler random variables. This kernel decays as a power law with exponent $\beta +1 \in (1,2]$. Several analytical results can be proved, in particular for the expected intensity of the point process and for the expected number of events of the counting process. These analytical results are used to validate algorithms that numerically invert the Laplace transform of the expected intensity as well as Monte Carlo simulations of the process. Finally, Monte Carlo simulations are used to derive the full distribution of the number of events. The algorithms used for this paper are available at {\tt https://github.com/habyarimanacassien/Fractional-Hawkes}.

A fractional Hawkes process II: Further characterization of the process

Scalas, Enrico;Polito, Federico
2023-01-01

Abstract

We characterize a Hawkes point process with kernel proportional to the probability density function of Mittag-Leffler random variables. This kernel decays as a power law with exponent $\beta +1 \in (1,2]$. Several analytical results can be proved, in particular for the expected intensity of the point process and for the expected number of events of the counting process. These analytical results are used to validate algorithms that numerically invert the Laplace transform of the expected intensity as well as Monte Carlo simulations of the process. Finally, Monte Carlo simulations are used to derive the full distribution of the number of events. The algorithms used for this paper are available at {\tt https://github.com/habyarimanacassien/Fractional-Hawkes}.
2023
615
128596
1
11
https://arxiv.org/pdf/2211.02583
Probability theory, stochastic processes
Habyarimana, Cassien; Aduda, Jane A.; Scalas, Enrico; Chen, Jing; Hawkes, Alan G.; Polito, Federico
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2318/1894174
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