L\'evy-areas of Ornstein-Uhlenbeck processes in Hilbert-spaces

TL;DR

This paper studies the existence, regularity, and stationarity of Lévy areas for Ornstein-Uhlenbeck processes driven by Hilbert-space-valued fractional Brownian motions with Hurst parameter between 1/3 and 1/2.

Contribution

It establishes the existence of Lévy areas for these processes, proves their Hölder continuity and stationarity, and provides approximation methods.

Findings

Lévy areas exist for the specified processes.

Lévy areas are Hölder continuous with high exponent.

Lévy areas are stationary processes.

Abstract

In this paper we investigate the existence and some useful properties of the L\'evy areas of Ornstein-Uhlenbeck processes associated to Hilbert-space-valued fractional Brownian-motions with Hurst parameter $H\in (1/3,1/2]$. We prove that this stochastic area has a H\"older-continuous version with sufficiently large H\"older-exponent and that can be approximated by smooth areas. In addition, we prove the stationarity of this area.