Affine pure-jump processes on positive Hilbert–Schmidt operators

We show the existence of a broad class of affine Markov processes on the cone of positive self-adjoint Hilbert–Schmidt operators. Such processes are well-suited as infinite-dimensional stochastic covariance models. The class of processes we consider is an infinite-dimensional analogue of the affine processes on the cone of positive semi-definite and symmetric matrices studied in Cuchiero et al. (2011). As in the finite-dimensional case, the processes we construct allow for a drift depending affine linearly on the state, as well as jumps governed by a jump measure that depends affine linearly on the state. The fact that the cone of positive self-adjoint Hilbert–Schmidt operators has empty interior calls for a new approach to proving existence: instead of using standard localization techniques, we employ the theory on generalized Feller semigroups introduced in Dörsek and Teichmann (2010) and further developed in Cuchiero and Teichmann (2020). Our approach requires a second moment condition on the jump measures involved, consequently, we obtain explicit formulas for the first and second moments of the affine process