The Fourier Cosine Method for Discrete Probability Distributions

TL;DR

This paper extends the Fourier cosine method to discrete probability distributions, providing convergence proofs, analyzing spectral filters, and demonstrating its efficiency in computational statistics and finance.

Contribution

It introduces a rigorous extension of the COS method for discrete distributions, with convergence analysis and practical applications.

Findings

Spectral filters improve convergence rates for the discrete COS method.

Numerical experiments confirm theoretical convergence and efficiency.

The method effectively computes discrete distributions when characteristic functions are known.

Abstract

We provide a rigorous convergence proof demonstrating that the well-known semi-analytical Fourier cosine (COS) formula for the inverse Fourier transform of continuous probability distributions can be extended to discrete probability distributions, with the help of spectral filters. We establish general convergence rates for these filters and further show that several classical spectral filters achieve convergence rates one order faster than previously recognized in the literature on the Gibbs phenomenon. Our numerical experiments corroborate the theoretical convergence results. Additionally, we illustrate the computational speed and accuracy of the discrete COS method with applications in computational statistics and quantitative finance. The theoretical and numerical results highlight the method's potential for solving problems involving discrete distributions, particularly when the characteristic function is known, allowing the discrete Fourier transform (DFT) to be bypassed.