Local H\"older regularity for set-indexed processes

TL;DR

This paper establishes a Kolmogorov-like H"older-continuity theorem for set-indexed stochastic processes, introduces various notions of H"older exponents, and applies these results to analyze the regularity of set-indexed fractional Brownian motion.

Contribution

It introduces a novel H"older-continuity theorem for set-indexed processes and compares different H"older exponents, with applications to Gaussian processes.

Findings

Proved a Kolmogorov-like H"older-continuity theorem for set-indexed processes.

Compared various definitions of H"older exponents for these processes.

Showed that the local regularity of set-indexed fractional Brownian motion equals the Hurst parameter almost surely.

Abstract

In this paper, we study the H\"older regularity of set-indexed stochastic processes defined in the framework of Ivanoff-Merzbach. The first key result is a Kolmogorov-like H\"older-continuity Theorem, whose novelty is illustrated on an example which could not have been treated with anterior tools. Increments for set-indexed processes are usually not simply written as $X_U-X_V$, hence we considered different notions of H\"older-continuity. Then, the localization of these properties leads to various definitions of H\"older exponents, which we compare to one another. In the case of Gaussian processes, almost sure values are proved for these exponents, uniformly along the sample paths. As an application, the local regularity of the set-indexed fractional Brownian motion is proved to be equal to the Hurst parameter uniformly, with probability one.