Sharp supremum and Hölder bounds for stochastic integrals indexed by a parameter

2024/09/20 by Cox, Sonja, van Winden, Joris · 2 citations
#60G42 #60G60 #60H05 #60H15 #FOS: Mathematics #Functional Analysis (math.FA) #Probability (math.PR)

paper · doi:10.48550/arxiv.2409.13615

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∈ 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 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.

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