  The HEat modulated Infinite DImensional Heston (HEIDIH) model and its
numerical approximation are introduced and analyzed. This model falls into the
general framework of infinite dimensional Heston stochastic volatility models
of (F.E. Benth, I.C. Simonsen '18), introduced for the pricing of forward
contracts. The HEIDIH model consists of a one-dimensional stochastic advection
equation coupled with a stochastic volatility process, defined as a
Cholesky-type decomposition of the tensor product of a Hilbert-space valued
Ornstein-Uhlenbeck process, the mild solution to the stochastic heat equation
on the real half-line. The advection and heat equations are driven by
independent space-time Gaussian processes which are white in time and colored
in space, with the latter covariance structure expressed by two different
kernels. First, a class of weight-stationary kernels are given, under which
regularity results for the HEIDIH model in fractional Sobolev spaces are
formulated. In particular, the class includes weighted Mat\'ern kernels.
Second, numerical approximation of the model is considered. An error
decomposition formula, pointwise in space and time, for a finite-difference
scheme is proven. For a special case, essentially sharp convergence rates are
obtained when this is combined with a fully discrete finite element
approximation of the stochastic heat equation. The analysis takes into account
a localization error, a pointwise-in-space finite element discretization error
and an error stemming from the noise being sampled pointwise in space. The
rates obtained in the analysis are higher than what would be obtained using a
standard Sobolev embedding technique. Numerical simulations illustrate the
results.
