Finite-rank approximation of affine processes on positive Hilbert-Schmidt operators

In this paper, we introduce a method for approximating affine processes on the cone of positive Hilbert-Schmidt operators using positive semi-definite matrix-valued affine processes. First, we identify a set of admissible parameters and a pair of associated operator-valued generalized Riccati equations that characterize the class of affine processes. We then show that certain finite-dimensional projections of these admissible parameters align with the parameters of a Galerkin approximation of the generalized Riccati equations. Leveraging the theory of matrix-valued affine processes, we show that these approximations are identifiable with a sequence of finite-rank operator-valued affine processes. By establishing convergence rates for the Galerkin approximation, we prove the weak convergence of this sequence of finite-rank operator-valued affine processes and provide convergence rates for their Laplace transforms. The introduced method not only offers a practical and efficient approximation scheme for operator-valued affine processes, but also introduces a novel proof for the existence of Hilbert-valued affine processes, in particular of càdlàg versions of these. An example of an affine operator-valued process with infinite variation and state-dependent jump intensity is highlighted. Beyond its theoretical implications, this paper offers valuable tools for the analysis and approximation of infinite-dimensional affine stochastic volatility models.