Absolute continuity of finite-dimensional distributions of Hermite processes via Malliavin calculus
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914056545239040 |
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| author | Loosveldt, Laurent Nachit, Yassine Nourdin, Ivan Tudor, Ciprian |
| author_facet | Loosveldt, Laurent Nachit, Yassine Nourdin, Ivan Tudor, Ciprian |
| contents | We investigate the existence of densities for finite-dimensional distributions of Hermite processes of order \(q \ge 1\) and self-similarity parameter \(H\in(\frac12,1)\). Whereas the Gaussian case \(q=1\) (fractional Brownian motion) is well understood, the non-Gaussian situation has not yet been settled. In this work, we extend the classical three-step approach used in the Gaussian case: factorization of the determinant into conditional terms, strong local nondeterminism, and non-degeneracy. We transport this strategy to the Hermite setting using Malliavin calculus. Specifically, we establish a determinant identity for the Malliavin matrix, prove strong local nondeterminism at the level of Malliavin derivatives, and apply the Bouleau-Hirsch criterion. Consequently, for any distinct times \(t_1,\dots,t_n\), the vector \((Z^{H,q}_{t_1},\dots,Z^{H,q}_{t_n})\) of a Hermite process admits a density with respect to the Lebesgue measure. Beyond the result itself, the main contribution is the methodology, which could extend to other non-Gaussian models. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_21169 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Absolute continuity of finite-dimensional distributions of Hermite processes via Malliavin calculus Loosveldt, Laurent Nachit, Yassine Nourdin, Ivan Tudor, Ciprian Probability 60G22, 60G18, 60H07, 60F05 We investigate the existence of densities for finite-dimensional distributions of Hermite processes of order \(q \ge 1\) and self-similarity parameter \(H\in(\frac12,1)\). Whereas the Gaussian case \(q=1\) (fractional Brownian motion) is well understood, the non-Gaussian situation has not yet been settled. In this work, we extend the classical three-step approach used in the Gaussian case: factorization of the determinant into conditional terms, strong local nondeterminism, and non-degeneracy. We transport this strategy to the Hermite setting using Malliavin calculus. Specifically, we establish a determinant identity for the Malliavin matrix, prove strong local nondeterminism at the level of Malliavin derivatives, and apply the Bouleau-Hirsch criterion. Consequently, for any distinct times \(t_1,\dots,t_n\), the vector \((Z^{H,q}_{t_1},\dots,Z^{H,q}_{t_n})\) of a Hermite process admits a density with respect to the Lebesgue measure. Beyond the result itself, the main contribution is the methodology, which could extend to other non-Gaussian models. |
| title | Absolute continuity of finite-dimensional distributions of Hermite processes via Malliavin calculus |
| topic | Probability 60G22, 60G18, 60H07, 60F05 |
| url | https://arxiv.org/abs/2509.21169 |