Infinite Dimensional Pathwise Volterra Processes Driven by Gaussian Noise -- Probabilistic Properties and Applications

TL;DR

This paper explores the mathematical properties of infinite dimensional Volterra processes driven by Gaussian noise, extending existing theories and applying them to rough path construction and fractional Ornstein-Uhlenbeck processes.

Contribution

It extends the Volterra sewing lemma to two dimensions, enabling new representations of covariance operators and applications to irregular Gaussian-driven processes.

Findings

Extended Volterra sewing lemma to two dimensions

Represented covariance operators as two-dimensional integrals

Constructed rough paths for irregular Gaussian processes

Abstract

We investigate the probabilistic and analytic properties of Volterra processes constructed as pathwise integrals of deterministic kernels with respect to the H\"older continuous trajectories of Hilbert-valued Gaussian processes. To this end, we extend the Volterra sewing lemma from \cite{HarangTindel} to the two dimensional case, in order to construct two dimensional operator-valued Volterra integrals of Young type. We prove that the covariance operator associated to infinite dimensional Volterra processes can be represented by such a two dimensional integral, which extends the current notion of representation for such covariance operators. We then discuss a series of applications of these results, including the construction of a rough path associated to a Volterra process driven by Gaussian noise with possibly irregular covariance structures, as well as a description of the irregular covariance structure arising from Gaussian processes time-shifted along irregular trajectories. Furthermore, we consider an infinite dimensional fractional Ornstein-Uhlenbeck process driven by Gaussian noise, which can be seen as an extension of the volatility model proposed by Rosenbaum et al. in \cite{ElEuchRosenbaum}.