Infinite dimensional pathwise Volterra processes driven by Gaussian noise – Probabilistic properties and applications –

Abstract

We investigate the probabilistic and analytic properties of Volterra processes constructed as pathwise integrals of deterministic kernels with respect to the Hölder 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].

Volterra equations driven by rough signals 3: Probabilistic construction of the Volterra rough path for fractional Brownian motions

“…One will therefore need to use new tools to handle this issue. We expect that techniques inspired by the results in [7], in combination with sewing techniques for Volterra covariance functions developed in [3], would prove useful to this aim. However, we leave this problem for future consideration.…”

Volterra equations driven by rough signals 2: Higher-order expansions

“…The Volterra sewing lemma, and most results relating to Volterra rough paths are however easily extended to general Volterra kernels k : ∆ 2 → L(E) for some Banach space E, by appropriate change of the bounds in 2.1, see e.g. [6,2] where the Volterra sewing lemma from [12] is readily applied in an infinite dimensional setting.…”

Volterra equations driven by rough signals 2: higher order expansions

“…The Volterra sewing lemma, and most results relating to Volterra rough paths are however easily extended to general Volterra kernels k : ∆ 2 → L(E) for some Banach space E, by appropriate change of the bounds in 2.1, see e.g. [9,6] where the Volterra sewing lemma from [1] is readily applied in an infinite dimensional setting.…”