  We introduce a flexible and tractable infinite-dimensional stochastic
volatility model. More specifically, we consider a Hilbert space valued
Ornstein-Uhlenbeck-type process, whose instantaneous covariance is given by a
pure-jump stochastic process taking values in the cone of positive self-adjoint
Hilbert-Schmidt operators. The tractability of our model lies in the fact that
the two processes involved are jointly affine, i.e., we show that their
characteristic function can be given explicitly in terms of the solutions to a
set of generalised Riccati equations. The flexibility lies in the fact that we
allow multiple modeling options for the instantaneous covariance process,
including state-dependent jump intensity. Infinite dimensional volatility
models arise e.g. when considering the dynamics of forward rate functions in
the Heath-Jarrow-Morton-Musiela modeling framework using the Filipovi\`c space.
In this setting we discuss various examples: an infinite-dimensional version of
the Barndorff-Nielsen-Shephard stochastic volatility model, as well as a model
involving self-exciting volatility.
