A Note on the Conditions for COS Convergence

ArXiv ID: 2512.02745 “View on arXiv”

Authors: Qinling Wang, Xiaoyu Shen, Fang Fang

Abstract

We study the truncation error of the COS method and give simple, verifiable conditions that guarantee convergence. In one dimension, COS is admissible when the density belongs to both L1 and L2 and has a finite weighted L2 moment of order strictly greater than one. We extend the result to multiple dimensions by requiring the moment order to exceed the dimension. These conditions enlarge the class of densities covered by previous analyses and include heavy-tailed distributions such as Student t with small degrees of freedom.

Keywords: COS method, truncation error, convergence conditions, heavy-tailed distributions, moment conditions, Quantitative Derivatives Pricing

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 1.0/10
• Quadrant: Lab Rats
• Why: The paper is highly theoretical, providing mathematical proofs and bounds for the convergence of the COS method using advanced functional analysis, placing it in the ‘Lab Rats’ quadrant. It lacks any empirical data, backtesting, or implementation details, with the primary output being analytical conditions for admissibility.

  flowchart TD
    A["Research Goal<br/>(What conditions guarantee<br/>COS method convergence?)"] --> B["Key Methodology<br/>(Mathematical error analysis)"]
    B --> C["Data/Inputs<br/>(Probability density functions)"]
    C --> D["Computational Process<br/>(Verify L1/L2 & finite weighted L2 moment<br/>moment order > dimension d)"]
    D --> E["Key Findings/Outcomes<br/>(Admissible densities include<br/>Student t & heavy-tailed distributions)"]