Hadamard fractional Brownian motion: path properties and Wiener integration
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915397741051904 |
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| author | Beghin, Luisa De Gregorio, Alessandro Mishura, Yuliya |
| author_facet | Beghin, Luisa De Gregorio, Alessandro Mishura, Yuliya |
| contents | The so-called Hadamard fractional Brownian motion, as defined in Beghin et al. (2025) by means of Hadamard fractional operators, is a Gaussian process which shares some properties with standard Brownian motion (such as the one-dimensional distribution). However, it also resembles the fractional Brownian motion in many other features as, for instance, self-similarity, long/short memory property, Wiener-integral representation. The logarithmic kernel in the Hadamard fractional Brownian motion represents a very specific and interesting aspect of this process. Our aim here is to analyze some properties of the process' trajectories (i.e. Hölder continuity, quasi-helix behavior, power variation, local nondeterminism) that are both interesting on their own and serve as a basis for the Wiener integration with respect to it. The respective integration is quite well developed, and the inverse representation is also constructed. We apply the derived ``multiplicative Sonine pairs'' to the treatment of the Reproducing Kernel Hilbert Space of the Hadamard fractional Brownian motion, and, as a result, we establish a law of iterated logarithm. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_13512 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hadamard fractional Brownian motion: path properties and Wiener integration Beghin, Luisa De Gregorio, Alessandro Mishura, Yuliya Probability Primary: 60G22. Secondary: 26A33, 60G15 The so-called Hadamard fractional Brownian motion, as defined in Beghin et al. (2025) by means of Hadamard fractional operators, is a Gaussian process which shares some properties with standard Brownian motion (such as the one-dimensional distribution). However, it also resembles the fractional Brownian motion in many other features as, for instance, self-similarity, long/short memory property, Wiener-integral representation. The logarithmic kernel in the Hadamard fractional Brownian motion represents a very specific and interesting aspect of this process. Our aim here is to analyze some properties of the process' trajectories (i.e. Hölder continuity, quasi-helix behavior, power variation, local nondeterminism) that are both interesting on their own and serve as a basis for the Wiener integration with respect to it. The respective integration is quite well developed, and the inverse representation is also constructed. We apply the derived ``multiplicative Sonine pairs'' to the treatment of the Reproducing Kernel Hilbert Space of the Hadamard fractional Brownian motion, and, as a result, we establish a law of iterated logarithm. |
| title | Hadamard fractional Brownian motion: path properties and Wiener integration |
| topic | Probability Primary: 60G22. Secondary: 26A33, 60G15 |
| url | https://arxiv.org/abs/2507.13512 |