Fractional interacting particle system: drift parameter estimation via Malliavin calculus
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917072113500160 |
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| author | Amorino, Chiara Nourdin, Ivan Shevchenko, Radomyra |
| author_facet | Amorino, Chiara Nourdin, Ivan Shevchenko, Radomyra |
| contents | We address the problem of estimating the drift parameter in a system of $N$ interacting particles driven by additive fractional Brownian motion of Hurst index \( H \geq 1/2 \). Considering continuous observation of the interacting particles over a fixed interval \([0, T]\), we examine the asymptotic regime as \( N \to \infty \). Our main tool is a random variable reminiscent of the least squares estimator but unobservable due to its reliance on the Skorohod integral. We demonstrate that this object is consistent and asymptotically normal by establishing a quantitative propagation of chaos for Malliavin derivatives, which holds for any \( H \in (0,1) \). Leveraging a connection between the divergence integral and the Young integral, we construct computable estimators of the drift parameter. These estimators are shown to be consistent and asymptotically Gaussian. Finally, a numerical study highlights the strong performance of the proposed estimators. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_06514 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fractional interacting particle system: drift parameter estimation via Malliavin calculus Amorino, Chiara Nourdin, Ivan Shevchenko, Radomyra Statistics Theory Probability We address the problem of estimating the drift parameter in a system of $N$ interacting particles driven by additive fractional Brownian motion of Hurst index \( H \geq 1/2 \). Considering continuous observation of the interacting particles over a fixed interval \([0, T]\), we examine the asymptotic regime as \( N \to \infty \). Our main tool is a random variable reminiscent of the least squares estimator but unobservable due to its reliance on the Skorohod integral. We demonstrate that this object is consistent and asymptotically normal by establishing a quantitative propagation of chaos for Malliavin derivatives, which holds for any \( H \in (0,1) \). Leveraging a connection between the divergence integral and the Young integral, we construct computable estimators of the drift parameter. These estimators are shown to be consistent and asymptotically Gaussian. Finally, a numerical study highlights the strong performance of the proposed estimators. |
| title | Fractional interacting particle system: drift parameter estimation via Malliavin calculus |
| topic | Statistics Theory Probability |
| url | https://arxiv.org/abs/2502.06514 |