A high-order recombination algorithm for weak approximation of stochastic differential equations

ArXiv ID: 2504.19717 “View on arXiv”

Authors: Syoiti Ninomiya, Yuji Shinozaki

Abstract

This paper presents an algorithm for applying the high-order recombination method, originally introduced by Lyons and Litterer in ``High-order recombination and an application to cubature on Wiener space’’ (Ann. Appl. Probab. 22(4):1301–1327, 2012), to practical problems in mathematical finance. A refined error analysis is provided, yielding a sharper condition for space partitioning. Based on this condition, a computationally feasible recursive partitioning algorithm is developed. Numerical examples are also included, demonstrating that the proposed algorithm effectively avoids the explosive growth in the cardinality of the support required to achieve high-order approximations.

Keywords: high-order recombination, cubature on Wiener space, recursive partitioning, error analysis, approximation methods, Mathematical Finance (Derivatives/Risk)

Complexity vs Empirical Score

• Math Complexity: 9.5/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats
• Why: The paper is heavily mathematical, featuring advanced stochastic calculus, measure theory, and recursive algorithms with extensive LaTeX derivations, but lacks empirical backtests, datasets, or implementation-heavy details, focusing instead on theoretical algorithmic development.

  flowchart TD
    A["Research Goal<br>Develop a computationally feasible<br>high-order recombination algorithm<br>for weak approximation of SDEs"] --> B["Key Methodology<br>Refined Error Analysis &<br>Recursive Partitioning Condition"]
    B --> C["Data/Input<br>Stochastic Differential Equations<br>in Mathematical Finance"]
    C --> D["Computational Process<br>High-Order Recombination &<br>Cardinality Control"]
    D --> E["Key Outcomes<br>Sharper partitioning criteria,<br>explosive growth avoidance,<br>high-order accuracy in finance"]