We consider stochastic partial differential equations appearing as Markovian lifts of matrix valued (affine) Volterra type processes from the point of view of the generalized Feller property (see e.g., \cite{doetei:10}). We introduce in particular Volterra Wishart processes with fractional kernels and values in the cone of positive semidefinite matrices. They are constructed from matrix products of infinite dimensional Ornstein Uhlenbeck processes whose state space are matrix valued measures. Parallel to that we also consider positive definite Volterra pure jump processes, giving rise to multivariate Hawkes type processes. We apply these affine covariance processes for multivariate (rough) volatility modeling and introduce a (rough) multivariate Volterra Heston type model.

中文翻译：

---

正半定仿射Volterra型过程的Markovian提升

从广义Feller性质的角度来看，我们认为随机偏微分方程以矩阵值（仿射）Volterra型过程的马尔可夫升序出现（例如，参见\ cite {doetei：10}）。我们特别介绍具有分数核和正半定矩阵圆锥中的值的Volterra Wishart过程。它们是由无限维Ornstein Uhlenbeck过程的矩阵乘积构造而成的，其状态空间是矩阵值的度量。与此平行的是，我们还考虑了正定的Volterra纯跳跃过程，从而产生了多元Hawkes型过程。我们将这些仿射协方差过程应用于多元（粗糙）波动率建模，并引入（粗糙）多元Volterra Heston类型模型。