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

TL;DR

This paper establishes sharp bounds for the supremum of stochastic convolutions in Banach spaces and applies these results to analyze the regularity of stochastic integrals, Ornstein-Uhlenbeck processes, and the parabolic Anderson model.

Contribution

It introduces sharp bounds for stochastic convolutions in 2-smooth Banach spaces and develops a theory of stochastic integration in H"older spaces for arbitrary bounded subsets of .

Findings

Sharp bounds for stochastic convolutions in Banach spaces.

Bounds on the modulus of continuity for stochastic integrals.

Applications to Ornstein-Uhlenbeck processes and the parabolic Anderson model.

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.