General convergence of the operator-discretization approximation for the characteristic function

Establish a general convergence theorem for the operator-discretization approximation of the characteristic function of the log-forward index under the Volterra Stein-Stein model with stochastic interest rates: prove that, for Volterra kernels G_nu satisfying Definition 2.1 and for complex u with real part in [0,1], the discrete approximation obtained by replacing the integral operators with their N×N discretizations (defining Psi_0^{u;N} and Phi_0^{u;N}) and the corresponding approximate characteristic function converges to the true characteristic function as the number of discretization points N tends to infinity.

Background

To enable Fourier pricing under general non-Markovian Volterra kernels, the paper derives a semi-explicit expression for the characteristic function of the log-forward index and proposes a practical approximation based on discretizing the associated integral operators on L^2([0,T]). This "operator-discretization" method replaces the kernel operator G_nu and the adjusted covariance operator with lower-triangular matrices, yielding a closed-form approximate characteristic function involving the discretized operator Psi_0^{u;N} and a Fredholm determinant proxy.

While this approximation is computationally efficient and numerically stable across examples (including fractional and shifted fractional kernels), the authors note that a general proof of convergence as the grid size N increases has not been established. In contrast, for completely monotone kernels, the paper provides an alternative multi-factor approximation with a theoretical convergence guarantee, highlighting the gap for the operator-discretization approach.