Ornstein–Uhlenbeck processes in Hilbert space with non-Gaussian stochastic volatility

Benth, Fred Espen; Rüdiger, Barbara; Süß, André

doi:10.1016/j.spa.2017.05.005

Cited by 21 publications

(49 citation statements)

References 23 publications

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“…Each stochastic process Y κ t is an OU process with exponential correlation (4), (18). The correlation of the process Z t is…”

Section: Resultsmentioning

confidence: 99%

“…In the derivation of (29), the first line follows from (8) and the second from the application of the rule of the sum of Gaussian variables reminding that the process Y κ t is Gaussian, see (18). The third line contains the multiplication and division by κ E[(Y κ t ) 2 ] and in the fourth τ eff is introduced and the parameter σ 0 is moved below the square root.…”

Section: Proof Of Theoremmentioning

confidence: 99%

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Centre-of-Mass Like Superposition of Ornstein–Uhlenbeck Processes: A Pathway to Non-Autonomous Stochastic Differential Equations and to Fractional Diffusion

D’Ovidio

Vitali

Sposini

et al. 2018

FCAA

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We consider an ensemble of Ornstein-Uhlenbeck processes featuring a population of relaxation times and a population of noise amplitudes that characterize the heterogeneity of the ensemble. We show that the centre-of-mass like variable corresponding to this ensemble is statistically equivalent to a process driven by a non-autonomous stochastic differential equation with timedependent drift and a white noise. In particular, the time scaling and the density function of such variable are driven by the population of timescales and of noise amplitudes, respectively. Moreover, we show that this variable is equivalent in distribution to a randomly-scaled Gaussian process, i.e., a process built by the product of a Gaussian process times a non-negative in- * dependent random variable. This last result establishes a connection with the so-called generalized gray Brownian motion and suggests application to model fractional anomalous diffusion in biological systems.

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“…Each stochastic process Y κ t is an OU process with exponential correlation (4), (18). The correlation of the process Z t is…”

Section: Resultsmentioning

confidence: 99%

Section: Proof Of Theoremmentioning

confidence: 99%

Centre-of-Mass Like Superposition of Ornstein–Uhlenbeck Processes: A Pathway to Non-Autonomous Stochastic Differential Equations and to Fractional Diffusion

D’Ovidio

Vitali

Sposini

et al. 2018

FCAA

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“…Let us return to a general separable Hilbert space H. Forward and futures prices can be realized as infinite dimensional stochastic processes, which call for operator-valued stochastic volatility models (see Benth, Rüdiger and Süss [7] and Benth and Krühner [6]).…”

Section: Rough Stochastic Volatility Modelsmentioning

confidence: 99%

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

Benth

Harang

2021

Electron. J. Probab.

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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].

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“…In this case, it would be natural to suppose L to be a Wiener process in L 2 (R), since the additional stochastic volatility process V will induce non-Gaussian distributed residuals. We leave the further discussion on stochastic volatility models in infinite di-mensional term structure models for future research (see however, Benth, Rüdiger, and Süss (2015) for a Hilbert-valued Ornstein-Uhlenbeck processes with stochastic volatility).…”

Section: Modeling Approach and Estimation Of The Noisementioning

confidence: 99%

A space-time random field model for electricity forward prices

Benth

Paraschiv

2018

Journal of Banking & Finance

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Stochastic models for forward electricity prices are of great relevance nowadays, given the major structural changes in the market due to the increase of renewable energy in the production mix. In this study, we derive a spatio-temporal dynamical model based on the Heath-Jarrow-Morton (HJM) approach under the Musiela parametrization, which ensures an arbitrage-free model for electricity forward prices. The model is fitted to a unique data set of historical price forward curves. As a particular feature of the model, we disentangle the temporal from spatial (maturity) effects on the dynamics of forward prices, and shed light on the statistical properties of risk premia, of the noise volatility term structure and of the spatio-temporal noise correlation structures. We find that the short-term risk premia oscillates around zero, but becomes negative in the long run. We identify the Samuelson effect in the volatility term structure and volatility bumps, explained by market fundamentals. Furthermore we find evidence for coloured noise and correlated residuals, which we model by a Hilbert space-valued normal inverse Gaussian Lévy process with a suitable covariance functional.

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Ornstein–Uhlenbeck processes in Hilbert space with non-Gaussian stochastic volatility

Cited by 21 publications

References 23 publications

Centre-of-Mass Like Superposition of Ornstein–Uhlenbeck Processes: A Pathway to Non-Autonomous Stochastic Differential Equations and to Fractional Diffusion

Centre-of-Mass Like Superposition of Ornstein–Uhlenbeck Processes: A Pathway to Non-Autonomous Stochastic Differential Equations and to Fractional Diffusion

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

A space-time random field model for electricity forward prices

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