Abstract
We investigate the probabilistic and analytic properties of Volterra processes constructed as pathwise integrals of deterministic kernels with respect to the Holder continuous trajectories of Hilbert-valued Gaussian processes. To this end, we extend the Volterra sewing lemma from [18] 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 [13].
Highlights
Volterra processes appear naturally in models with non-local features
We will investigate the analytic and probabilistic properties of Volterra processes constructed as pathwise integrals of a kernel K against a Gaussian process W
Through a consideration of the characteristic functional associated to the Volterra processes (1.1), we show that when Q = QW is of sufficient regularity, the covariance operator QX associated the Volterra process X from (1.1) is given by Qin (1.5)
Summary
Volterra processes appear naturally in models with non-local features. In this article, we will investigate the analytic and probabilistic properties of Volterra processes constructed as pathwise integrals of a kernel K against a Gaussian process W. The main goal of this article is to extend the sufficient conditions for construction of the covariance operator on the form of (1.2) to the case when W is an infinite dimensional stochastic process and QW is possibly nowhere differentiable in both variables To this end, we start by giving a pathwise description of the Volterra process X stated in (1.1). In the end we discuss several application areas of our results, including an analysis of the covariance structure arising from general Gaussian Volterra iterated processes, the construction of the rough path associated to Volterra processes driven by Gaussian processes with irregular covariance structures, as well as a representation of the covariance structure of certain linear fractional stochastic differential equations of Ornstein-Uhlenbeck type in Hilbert space. In the current article, we do not impose any further structure on the covariance operator, other than regularity to keep it as general as possible, which in the end will prove useful in applications
Sign-in/Register to view highlights & summary 
Published Version (
Free)
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call