Affine pure-jump processes on positive Hilbert-Schmidt operators

Abstract: 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.