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Sharp supremum and H\"older bounds for stochastic integrals indexed by a parameter

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arxiv 2409.13615 v1 pith:BYJGXX4C submitted 2024-09-20 math.PR math.FA

Sonja Cox , Joris van Winden This is my paper · ORCID
classification math.PRmath.FA
keywords boundsstochasticcontinuitymodulussharpsupremumcountablyindexed
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abstract

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 H\"older spaces on arbitrary bounded subsets of $\mathbb{R}^d$. This is done by relating the (generalized) H\"older-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.

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