Sharp supremum and Hölder bounds for stochastic integrals indexed by a parameter
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916403770032128 |
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| author | Cox, Sonja van Winden, Joris |
| author_facet | Cox, Sonja van Winden, Joris |
| contents | We provide sharp bounds for the supremum of countably many stochastic convolutions taking values in a 2-smooth Banach space. As a consequence, we obtain sharp bounds on the modulus of continuity of a family of stochastic integrals indexed by parameter $x\in M$, where $M$ is a metric space with finite doubling dimension. In particular, we obtain a theory of stochastic integration in Hölder spaces on arbitrary bounded subsets of $\mathbb{R}^d$. This is done by relating the (generalized) Hölder-seminorm associated with a modulus of continuity to a supremum over countably many variables, using a Kolmogorov-type chaining argument. We provide two applications of our results: first, we show long-term bounds for Ornstein-Uhlenbeck processes, and second, we derive novel results regarding the modulus of continuity of the parabolic Anderson model. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_13615 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sharp supremum and Hölder bounds for stochastic integrals indexed by a parameter Cox, Sonja van Winden, Joris Probability Functional Analysis 60H05, 60G60, 60H15, 60G42 We provide sharp bounds for the supremum of countably many stochastic convolutions taking values in a 2-smooth Banach space. As a consequence, we obtain sharp bounds on the modulus of continuity of a family of stochastic integrals indexed by parameter $x\in M$, where $M$ is a metric space with finite doubling dimension. In particular, we obtain a theory of stochastic integration in Hölder spaces on arbitrary bounded subsets of $\mathbb{R}^d$. This is done by relating the (generalized) Hölder-seminorm associated with a modulus of continuity to a supremum over countably many variables, using a Kolmogorov-type chaining argument. We provide two applications of our results: first, we show long-term bounds for Ornstein-Uhlenbeck processes, and second, we derive novel results regarding the modulus of continuity of the parabolic Anderson model. |
| title | Sharp supremum and Hölder bounds for stochastic integrals indexed by a parameter |
| topic | Probability Functional Analysis 60H05, 60G60, 60H15, 60G42 |
| url | https://arxiv.org/abs/2409.13615 |