Paper: arXiv 2512.24714

Authors: Xiang Gao, Cody Hyndman

Abstract

We first review the convolution fast-Fourier-transform (CFFT) approach for the numerical solution of backward stochastic differential equations (BSDEs) introduced in (Hyndman and Oyono Ngou, 2017). We then propose a method for improving the boundary errors obtained when valuing options using this approach. We modify the damping and shifting schemes used in the original formulation, which transforms the target function into a bounded periodic function so that Fourier transforms can be applied successfully. Time-dependent shifting reduces boundary error significantly. We present numerical results for our implementation and provide a detailed error analysis showing the improved accuracy and convergence of the modified convolution method.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 2.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper involves advanced stochastic calculus, partial differential equations, and Fourier analysis with detailed error bounds and convergence proofs. The empirical component is limited to numerical illustrations and theoretical error analysis without real-world data, backtests, or implementation details.

Research Flowchart

  flowchart TD
  A["Research Goal<br>Improve boundary error control in<br>CFFT method for BSDEs"] --> B["Methodology<br>Modify damping & shifting schemes<br>Time-dependent shifting for boundary error"]
  B --> C["Computational Process<br>Implement modified CFFT with<br>numerical experiments"]
  C --> D["Key Findings<br>Significantly reduced boundary errors<br>Faster convergence & higher accuracy<br>Validated with error analysis"]