Stochastic covariance models in Hilbert spaces with jumps

In this thesis we provide the mathematical foundations for two novel classes of operator-valued stochastic processes with jumps that can be used as models for the instantaneous covariance process in stochastic covariance models in finite-and infinite-dimensional Hilbert spaces. The natural state-space for such processes is the cone of positive self-adjoint Hilbert-Schmidt operators, which is the natural infinite-dimensional version of the cone of positive semi-definite and symmetric matrices. The first class that we study, is the class of affine processes on (infinite-dimensional) positive Hilbert-Schmidt operators. The second is the class of positive semi-definite matrix-valued MCARMA processes.