Lévy processes as weak limits of rough Heston models
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918413757054976 |
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| author | Bondi, Alessandro Forde, Martin |
| author_facet | Bondi, Alessandro Forde, Martin |
| contents | We show weak convergence of the time-$t$ marginals for the integrated variance in a re-scaled rough Heston model to an Inverse Gaussian Lévy process. This shows we can obtain such a limit without having to impose that the true Hurst exponent $H$ for the model is $\frac{1}{2}$ as in [Abi Jaber, & De Carvalho, 2024], or that $H\searrow -\frac{1}{2}$ as in [Abi Jaber, Attal, & Rosenbaum, 2025], so the result potentially has increased financial relevance. We later extend the analysis to the case where $V$ has jumps, showing weak convergence of the finite-dimensional distributions of the integrated variance to a deterministic time-change of the first-passage time process to lower barriers for a more general class of spectrally positive Lévy processes. This convergence result is then strengthened to a functional setting, namely on the space of càdlàg functions on the non-negative half-line endowed with the $M_1$ topology. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_14835 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Lévy processes as weak limits of rough Heston models Bondi, Alessandro Forde, Martin Probability 60H20, 45D05, 60F05, 60G22 We show weak convergence of the time-$t$ marginals for the integrated variance in a re-scaled rough Heston model to an Inverse Gaussian Lévy process. This shows we can obtain such a limit without having to impose that the true Hurst exponent $H$ for the model is $\frac{1}{2}$ as in [Abi Jaber, & De Carvalho, 2024], or that $H\searrow -\frac{1}{2}$ as in [Abi Jaber, Attal, & Rosenbaum, 2025], so the result potentially has increased financial relevance. We later extend the analysis to the case where $V$ has jumps, showing weak convergence of the finite-dimensional distributions of the integrated variance to a deterministic time-change of the first-passage time process to lower barriers for a more general class of spectrally positive Lévy processes. This convergence result is then strengthened to a functional setting, namely on the space of càdlàg functions on the non-negative half-line endowed with the $M_1$ topology. |
| title | Lévy processes as weak limits of rough Heston models |
| topic | Probability 60H20, 45D05, 60F05, 60G22 |
| url | https://arxiv.org/abs/2508.14835 |