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

TL;DR

This paper extends Ornstein-Uhlenbeck processes to Hilbert spaces with non-Gaussian Levy noise, providing explicit characteristic functions and applications to commodity futures pricing.

Contribution

It introduces a non-Gaussian operator-valued stochastic volatility model with explicit characteristic functions and analytical solutions under certain conditions.

Findings

Explicit characteristic function for the non-Gaussian volatility process

Analytical expression for the characteristic functional of the perturbed OU process

Application to futures prices in commodity markets

Abstract

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