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We propose a non-Gaussian operator-valued extension of the Barndorff-Nielsenand Shephard stochastic volatility dynamics, defined as the square-root of anoperator-valued Ornstein-Uhlenbeck process with Levy noise and bounded drift.We derive conditions for the positive definiteness of the Ornstein-Uhlenbeckprocess, where in particular we must restrict to operator-valued Levy processeswith "non-decreasing paths". It turns out that the volatility model allows foran explicit calculation of its characteristic function, showing an affinestructure. We introduce another Hilbert space-valued Ornstein-Uhlenbeck processwith Wiener noise perturbed by this class of stochastic volatility dynamics.Under a strong commutativity condition between the covariance operator of theWiener process and the stochastic volatility, we can derive an analyticalexpression for the characteristic functional of the Ornstein-Uhlenbeck processperturbed by stochastic volatility if the noises are independent. The case ofoperator-valued compound Poisson processes as driving noise in the volatilityis discussed as a particular example of interest. We apply our results tofutures prices in commodity markets, where we discuss our proposed stochasticvolatility model in light of ambit fields.

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