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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2503.16985 |
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| _version_ | 1866909547352817664 |
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| author | Jaber, Eduardo Abi Attal, Elie Rosenbaum, Mathieu |
| author_facet | Jaber, Eduardo Abi Attal, Elie Rosenbaum, Mathieu |
| contents | We investigate the weak limit of the hyper-rough square-root process as the Hurst index $H$ goes to $-1/2\,$. This limit corresponds to the fractional kernel $t^{H - 1 / 2}$ losing integrability. We establish the joint convergence of the couple $(X, M)\,$, where $X$ is the hyper-rough process and $M$ the associated martingale, to a fully correlated Inverse Gaussian Lévy jump process. This unveils the existence of a continuum between hyper-rough continuous models and jump processes, as a function of the Hurst index. Since we prove a convergence of continuous to discontinuous processes, the usual Skorokhod $J_1$ topology is not suitable for our problem. Instead, we obtain the weak convergence in the Skorokhod $M_1$ topology for $X$ and in the non-Skorokhod $S$ topology for $M$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_16985 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | From Hyper Roughness to Jumps as $H \to -1/2$ Jaber, Eduardo Abi Attal, Elie Rosenbaum, Mathieu Probability We investigate the weak limit of the hyper-rough square-root process as the Hurst index $H$ goes to $-1/2\,$. This limit corresponds to the fractional kernel $t^{H - 1 / 2}$ losing integrability. We establish the joint convergence of the couple $(X, M)\,$, where $X$ is the hyper-rough process and $M$ the associated martingale, to a fully correlated Inverse Gaussian Lévy jump process. This unveils the existence of a continuum between hyper-rough continuous models and jump processes, as a function of the Hurst index. Since we prove a convergence of continuous to discontinuous processes, the usual Skorokhod $J_1$ topology is not suitable for our problem. Instead, we obtain the weak convergence in the Skorokhod $M_1$ topology for $X$ and in the non-Skorokhod $S$ topology for $M$. |
| title | From Hyper Roughness to Jumps as $H \to -1/2$ |
| topic | Probability |
| url | https://arxiv.org/abs/2503.16985 |