Tempered Fractional Hawkes Process and Its Generalization
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913352682307584 |
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| author | Gupta, Neha Maheshwari, Aditya |
| author_facet | Gupta, Neha Maheshwari, Aditya |
| contents | Hawkes process (HP) is a point process with a conditionally dependent intensity function. This paper defines the tempered fractional Hawkes process (TFHP) by time-changing the HP with an inverse tempered stable subordinator. We obtained results that generalize the fractional Hawkes process defined in Hainaut (2020) to a tempered version which has \textit{semi-heavy tailed} decay. We derive the mean, the variance, covariance and the governing fractional difference-differential equations of the TFHP. Additionally, we introduce the generalized fractional Hawkes process (GFHP) by time-changing the HP with the inverse Lévy subordinator. This definition encompasses all potential (inverse Lévy) time changes as specific instances. We also explore the distributional characteristics and the governing difference-differential equation of the one-dimensional distribution for the GFHP. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_09966 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Tempered Fractional Hawkes Process and Its Generalization Gupta, Neha Maheshwari, Aditya Probability 60G22, 60G51, 60G55 Hawkes process (HP) is a point process with a conditionally dependent intensity function. This paper defines the tempered fractional Hawkes process (TFHP) by time-changing the HP with an inverse tempered stable subordinator. We obtained results that generalize the fractional Hawkes process defined in Hainaut (2020) to a tempered version which has \textit{semi-heavy tailed} decay. We derive the mean, the variance, covariance and the governing fractional difference-differential equations of the TFHP. Additionally, we introduce the generalized fractional Hawkes process (GFHP) by time-changing the HP with the inverse Lévy subordinator. This definition encompasses all potential (inverse Lévy) time changes as specific instances. We also explore the distributional characteristics and the governing difference-differential equation of the one-dimensional distribution for the GFHP. |
| title | Tempered Fractional Hawkes Process and Its Generalization |
| topic | Probability 60G22, 60G51, 60G55 |
| url | https://arxiv.org/abs/2405.09966 |