A fast and accurate numerical method for the symmetric L\'evy processes based on the Fourier transform and sinc-Gauss sampling formula

TL;DR

This paper introduces a fast, accurate numerical method using Fourier transform and sinc-Gauss sampling for solving Kolmogorov forward equations of symmetric Lévy processes, achieving exponential convergence and computational efficiency.

Contribution

It combines Fourier transform formulas with FFT and sinc-Gauss sampling to improve speed and accuracy in solving Lévy process equations without special treatment for non-smooth initial conditions.

Findings

Method achieves exponential convergence for smooth solutions.

Computational times align with theoretical estimates.

No special handling needed for non-smooth initial conditions.

Abstract

In this paper, we propose a fast and accurate numerical method based on Fourier transform to solve Kolmogorov forward equations of symmetric scalar L\'evy processes. The method is based on the accurate numerical formulas for Fourier transform proposed by Ooura. These formulas are combined with nonuniform fast Fourier transform (FFT) and fractional FFT to speed up the numerical computations. Moreover, we propose a formula for numerical indefinite integration on equispaced grids as a component of the method. The proposed integration formula is based on the sinc-Gauss sampling formula, which is a function approximation formula. This integration formula is also combined with the FFT. Therefore, all steps of the proposed method are executed using the FFT and its variants. The proposed method allows us to be free from some special treatments for a non-smooth initial condition and numerical time integration. The numerical solutions obtained by the proposed method appeared to be exponentially convergent on the interval if the corresponding exact solutions do not have sharp cusps. Furthermore, the real computational times are approximately consistent with the theoretical estimates.