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Infinite Dimensional Pathwise Volterra Processes Driven by Gaussian Noise -- Probabilistic Properties and Applications
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}.Comment: 38 page
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