Long time Hurst regularity of fractional SDEs and their ergodic means

TL;DR

This paper extends the regularity properties of fractional Brownian motion and its SDE solutions to the entire positive time axis, establishing uniform H"older continuity in the Hurst parameter and analyzing ergodic means for statistical applications.

Contribution

It generalizes the joint H"older continuity of fractional Brownian motion to all positive times and studies the regularity of solutions and ergodic means of fractional SDEs with contractive drift.

Findings

Fractional Brownian motion is almost surely jointly H"older continuous in time and Lipschitz in H on the entire positive axis.

Solutions to fractional SDEs with contractive drift are almost surely H"older continuous in H, uniformly over time.

Invariant measures are sensitive to changes in the Hurst parameter H.

Abstract

The fractional Brownian motion can be considered as a Gaussian field indexed by $(t,H)\in {\mathbb{R}_{+}\times (0,1)}$, where $H$ is the Hurst parameter. On compact time intervals, it is known to be almost surely jointly H\"older continuous in time and Lipschitz continuous in $H$. First, we extend this result to the whole time interval $\mathbb{R}_{+}$ and consider both simple and rectangular increments. Then we consider SDEs driven by fractional Brownian motion with contractive drift. The solutions and their ergodic means are proven to be almost surely H\"older continuous in $H$, uniformly in time. This result is used in a separate work for statistical applications. We also deduce a sensibility result of the invariant measure in $H$. The proofs are based on variance estimates of the increments of the fractional Brownian motion and fractional Ornstein-Uhlenbeck processes, multiparameter versions of the Garsia-Rodemich-Rumsey lemma and a combinatorial argument to estimate the expectation of a product of Gaussian variables.