The HEat modulated Infinite DImensional Heston (HEIDEH) model and its numerical approximation are introduced and analysed. This is a special case of the infinite dimensional Heston stochastic volatility model of (F.E. Benth, I. C. Simonsen '18). Therein, the authors consider a potential model for risk-neutral forward prices of commodity-delivering contracts. The 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. In this work the Ornstein-Uhlenbeck process is specified to be the solution to a stochastic heat equation and the resulting HEIDEH model is studied in a fractional Sobolev space setting.
In the talk, a description and motivation of the model will be given. Then, a class of covariance kernels are described that give rise to admissible Q-Wiener processes. Regularity of the model is discussed under this class of kernels. Finally, an approximation for a special case based on a combination of an explicit finite difference scheme, a finite element method, the backward Euler scheme and the circulant embedding method is presented. Convergence rates are derived with the error measured pointwise, in a mean square sense, in time and space. The resulting rates are higher than what can be obtained from a standard Sobolev embedding technique.
This is joint work with Fred Espen Benth, Gabriel Lord and Giulia Di Nunno.