Convergence of nonhomogeneous Hawkes processes and Feller random measures
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910108393406464 |
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| author | Pace, Tristan Zitkovic, Gordan |
| author_facet | Pace, Tristan Zitkovic, Gordan |
| contents | We consider a sequence of Hawkes processes whose excitation measures may depend on the generation, and study its scaling limits in the near-unstable limiting regime. The limiting random measures, characterized via a nonlinear convolutional equation, form a family parameterized by a pair consisting of a locally finite measure and a geometrically infinitely divisible probability distribution on the positive real line. These measures can be interpreted as generalizations of the Feller diffusion and fractional Feller (CIR) processes, but also allow for a "driving noise" associated with general L\' evy-type operators of order at most $1$, including fractional derivatives of any order $α>0$ (formally corresponding to possibly negative Hurst parameters). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_15999 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Convergence of nonhomogeneous Hawkes processes and Feller random measures Pace, Tristan Zitkovic, Gordan Probability 60F17, 60G55, 60G57, 45D05 We consider a sequence of Hawkes processes whose excitation measures may depend on the generation, and study its scaling limits in the near-unstable limiting regime. The limiting random measures, characterized via a nonlinear convolutional equation, form a family parameterized by a pair consisting of a locally finite measure and a geometrically infinitely divisible probability distribution on the positive real line. These measures can be interpreted as generalizations of the Feller diffusion and fractional Feller (CIR) processes, but also allow for a "driving noise" associated with general L\' evy-type operators of order at most $1$, including fractional derivatives of any order $α>0$ (formally corresponding to possibly negative Hurst parameters). |
| title | Convergence of nonhomogeneous Hawkes processes and Feller random measures |
| topic | Probability 60F17, 60G55, 60G57, 45D05 |
| url | https://arxiv.org/abs/2412.15999 |