Pathwise Itô isometry for scaled quadratic variation
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | |
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| _version_ | 1866912485653610496 |
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| author | Bhar, Suprio Das, Purba Sarkar, Barun |
| author_facet | Bhar, Suprio Das, Purba Sarkar, Barun |
| contents | The concept of scaled quadratic variation was originally introduced by E. Gladyshev in 1961 in the context of Gaussian processes, where it was defined as the limit of the covariance of the underlying Gaussian process. In this paper, we extend this notion beyond the Gaussian framework for any real-valued continuous function by formulating it in a pathwise manner along a given sequence of partitions. We demonstrate that, for classical Gaussian processes such as fractional Brownian motion, this pathwise definition coincides with the traditional one up to a constant factor. Furthermore, we establish that the scaled quadratic variation is invariant under smooth transformations and satisfies a pathwise Itô isometry-type result, derived without relying on any expectation arguments. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_18290 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Pathwise Itô isometry for scaled quadratic variation Bhar, Suprio Das, Purba Sarkar, Barun Probability 60G17, 60G22, 60H20, 60L99 The concept of scaled quadratic variation was originally introduced by E. Gladyshev in 1961 in the context of Gaussian processes, where it was defined as the limit of the covariance of the underlying Gaussian process. In this paper, we extend this notion beyond the Gaussian framework for any real-valued continuous function by formulating it in a pathwise manner along a given sequence of partitions. We demonstrate that, for classical Gaussian processes such as fractional Brownian motion, this pathwise definition coincides with the traditional one up to a constant factor. Furthermore, we establish that the scaled quadratic variation is invariant under smooth transformations and satisfies a pathwise Itô isometry-type result, derived without relying on any expectation arguments. |
| title | Pathwise Itô isometry for scaled quadratic variation |
| topic | Probability 60G17, 60G22, 60H20, 60L99 |
| url | https://arxiv.org/abs/2504.18290 |