Markovian lifts of positive semidefinite affine Volterra-type processes

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., Dorsek and Teichmann in A semigroup point of view on splitting schemes for stochastic (partial) differential equations, 2010. ). 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 is the set of 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.