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.

翻译：本文引入并分析了热调制无穷维Heston（HEIDIH）模型及其数值逼近方法。该模型属于（F.E. Benth, I.C. Simonsen '18）为远期合约定价提出的无穷维Heston随机波动率模型的一般框架。HEIDIH模型由一维随机平流方程与随机波动率过程耦合构成，其中波动率过程定义为希尔伯特空间值Ornstein-Uhlenbeck过程的张量积的Cholesky型分解，该过程是实半线上随机热方程的温和解。平流方程和热方程由独立的空间-时间高斯过程驱动，这些过程在时间上为白噪声，在空间上具有颜色依赖性，且后者的协方差结构通过两种不同核函数表达。首先，给出了一类权重平稳核函数，在此条件下建立了HEIDIH模型在分数阶Sobolev空间中的正则性结果。特别地，该类核函数包含加权Matérn核函数。其次，研究了模型的数值逼近方法。证明了有限差分格式在空间和时间点态上的误差分解公式。在特例情形下，当该方法与随机热方程的全离散有限元逼近相结合时，获得了本质最优的收敛速率。该分析考虑了局部化误差、空间点态有限元离散误差以及由噪声空间点态采样引起的误差。分析得到的收敛速率高于采用标准Sobolev嵌入技术所得的结果。数值仿真验证了理论结果。