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2021 Infinite dimensional pathwise Volterra processes driven by Gaussian noise – Probabilistic properties and applications –
Fred Espen Benth, Fabian A. Harang
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Electron. J. Probab. 26: 1-42 (2021). DOI: 10.1214/21-EJP683


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].

Funding Statement

The authors gratefully acknowledge financial support from the STORM project 274410, funded by the Research Council of Norway.


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Fred Espen Benth. Fabian A. Harang. "Infinite dimensional pathwise Volterra processes driven by Gaussian noise – Probabilistic properties and applications –." Electron. J. Probab. 26 1 - 42, 2021.


Received: 29 May 2020; Accepted: 7 August 2021; Published: 2021
First available in Project Euclid: 8 September 2021

arXiv: 2005.14460
Digital Object Identifier: 10.1214/21-EJP683

Primary: 34A12 , 45D05 , 60H05 , 60H20

Keywords: Covariance operator , Fractional differential equations , Gaussian processes , Hilbert space , infinite dimensional stochastic analysis , rough path integration , rough volatility models , Volterra integral equations

Vol.26 • 2021
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