  We show the existence of a broad class of affine Markov processes in the cone
of positive self-adjoint Hilbert-Schmidt operators. Such processes are
well-suited as infinite dimensional stochastic volatility models. The class of
processes we consider is an infinite dimensional analogue of the affine
processes in the space of positive semi-definite and symmetric matrices studied
in Cuchiero et al. [Ann. Appl. Probab. 21 (2011) 397-463]. As in the finite
dimensional case, the processes we construct allow for a drift depending affine
linearly on the state, as well jumps governed by a jump measure that depends
affine linearly on the state. However, because the infinite-dimensional cone of
positive self-adjoint Hilbert-Schmidt operators has empty interior, we do not
consider a diffusion term. This empty interior also demands a new approach to
proving existence: instead of using standard localisation techniques, we employ
the theory on generalized Feller semigroups introduced in D\"orsek and
Teichmann [arXiv 2010] and further developed in Cuchiero and Teichmann [Journal
of Evolution Equations (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.
