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

TL;DR

This paper introduces a practical method for approximating affine processes on positive Hilbert-Schmidt operators using finite-rank matrix processes, with proven convergence and error bounds, applicable in mathematical finance.

Contribution

It develops a novel finite-rank approximation scheme for operator-valued affine processes, including a new existence proof and convergence analysis.

Findings

Finite-rank affine processes converge weakly to the target process.

Error bounds for Laplace transform approximations are established.

Applicable to infinite-dimensional stochastic covariance models in finance.

Abstract

In this article, we present a method for approximating affine processes on the cone of positive Hilbert-Schmidt operators using matrix-valued affine processes. By leveraging results from the theory on affine processes with values in the cone of symmetric and positive semi-definite matrices, we construct sequences of finite-rank operator-valued affine processes that converge weakly to the target processes and provide convergence rates for their Laplace transforms using Galerkin approximations of the associated operator-valued generalized Riccati equations. This article not only offers a practical approximation scheme for operator-valued affine processes with error bounds that hold uniformly in time, but also provides a novel existence proof for this class of affine processes with c\`adl\`ag paths, including affine pure-jump processes with infinite variation and state-dependent jump intensities. In addition to the theoretical significance, the results of this paper provide useful tools for analyzing and approximating infinite-dimensional affine stochastic covariance models that were recently introduced in mathematical finance.