A new architecture of high-order deep neural networks that learn martingales

ArXiv ID: 2505.03789 “View on arXiv”

Authors: Syoiti Ninomiya, Yuming Ma

Abstract

A new deep-learning neural network architecture based on high-order weak approximation algorithms for stochastic differential equations (SDEs) is proposed. The architecture enables the efficient learning of martingales by deep learning models. The behaviour of deep neural networks based on this architecture, when applied to the problem of pricing financial derivatives, is also examined. The core of this new architecture lies in the high-order weak approximation algorithms of the explicit Runge–Kutta type, wherein the approximation is realised solely through iterative compositions and linear combinations of vector fields of the target SDEs.

Keywords: Deep Learning, Stochastic Differential Equations, High-Order Weak Approximation, Runge-Kutta, Option Pricing, Equities

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 4.0/10
• Quadrant: Lab Rats
• Why: The paper proposes a novel neural network architecture inspired by high-order weak approximation algorithms for SDEs, requiring advanced stochastic calculus and numerical analysis (e.g., Stratonovich integrals, Runge–Kutta methods, vector fields). While the derivative pricing application suggests practical relevance, the summary lacks implementation details, backtests, or empirical validation data, focusing instead on theoretical framework and methodology.

  flowchart TD
    A["Research Goal: <br>Learn Martingales via Deep Learning"] --> B{"Data: Financial SDEs &<br>Derivative Pricing Problems"}
    B --> C["Methodology: High-Order RK Architecture"]
    C --> D["Process: Iterative Composition &<br>Linear Combination of Vector Fields"]
    D --> E["Outcome: Efficient Pricing of<br>Financial Derivatives"]
    E --> F["Key Finding: High-Order Weak Approximation<br>enables accurate SDE learning"]