Mean-Field Limits for Nearly Unstable Hawkes Processes
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866909462759997440 |
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| author | Szymanski, Grégoire Xu, Wei |
| author_facet | Szymanski, Grégoire Xu, Wei |
| contents | In this paper, we establish general scaling limits for nearly unstable Hawkes processes in a mean-field regime by extending the method introduced by Jaisson and Rosenbaum. Under a mild asymptotic criticality condition on the self-exciting kernels $\{ϕ^n\}$, specifically $\|ϕ^n\|_{L^1} \to 1$, we first show that the scaling limits of these Hawkes processes are necessarily stochastic Volterra diffusions of affine type. Moreover, we establish a propagation of chaos result for Hawkes systems with mean-field interactions, highlighting three distinct regimes for the limiting processes, which depend on the asymptotics of $n(1-\|ϕ^n\|_{L^1})^2$. These results provide a significant generalization of the findings by Delattre, Fournier and Hoffmann. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_11648 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Mean-Field Limits for Nearly Unstable Hawkes Processes Szymanski, Grégoire Xu, Wei Probability Statistical Finance 60F05, 60G55, 60G22, 60F17, 60G57 In this paper, we establish general scaling limits for nearly unstable Hawkes processes in a mean-field regime by extending the method introduced by Jaisson and Rosenbaum. Under a mild asymptotic criticality condition on the self-exciting kernels $\{ϕ^n\}$, specifically $\|ϕ^n\|_{L^1} \to 1$, we first show that the scaling limits of these Hawkes processes are necessarily stochastic Volterra diffusions of affine type. Moreover, we establish a propagation of chaos result for Hawkes systems with mean-field interactions, highlighting three distinct regimes for the limiting processes, which depend on the asymptotics of $n(1-\|ϕ^n\|_{L^1})^2$. These results provide a significant generalization of the findings by Delattre, Fournier and Hoffmann. |
| title | Mean-Field Limits for Nearly Unstable Hawkes Processes |
| topic | Probability Statistical Finance 60F05, 60G55, 60G22, 60F17, 60G57 |
| url | https://arxiv.org/abs/2501.11648 |