Numerical methods for two-dimensional G-heat equation

Paper: arXiv 2503.02395

Authors: Z. T. Pei, X. Y. Yue, X. T. Zheng

Abstract

The G-expectation is a sublinear expectation. It is an important tool for pricing financial products and managing risk thanks to its ability to deal with model uncertainty. The problem is how to efficiently quantify it since the commonly used Monte Carlo method does not work. Fortunately, the expectation of a G-normal random variable can be linked to the viscosity solution of a fully nonlinear G-heat equation. In this paper, we propose a novel numerical scheme for the two-dimensional G-heat equation and pay more attention to the case that there exists uncertainty on the correlationship, especially to the case that the correlationship ranges from negative to positive. The scheme is monotonic, stable, and convergent. The numerical tests show that the scheme is highly efficient.

Complexity vs Empirical Score

• Math Complexity: 8.0/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical formulation for solving the two-dimensional G-heat equation under correlation sign uncertainty using finite difference schemes, viscosity solutions, and proof of M-matrix stability. However, it is primarily a mathematical/numerical analysis paper with minimal empirical or market data backtesting presented in the excerpt.

Research Flowchart

  flowchart TD
    A["Research Goal:<br>Efficiently quantify G-expectation under correlation uncertainty"] --> B["Inputs:<br>2D G-heat equation & variable correlation ranges (- to +)"]
    B --> C["Develop Numerical Scheme:<br>Monotonic, stable, and convergent finite difference method"]
    C --> D["Computational Engine:<br>Solve fully nonlinear PDE for viscosity solution"]
    D --> E["Key Outcomes:<br>Highly efficient scheme for pricing derivatives under model uncertainty"]