Symmetry classification and invariant solutions of the classical geometric mean reversion process

Paper: arXiv 2504.13094

Authors: Jin Zhang, Dapeng Gao

Abstract

Based on the Lie symmetry method, we investigate a Feynman-Kac formula for the classical geometric mean reversion process, which effectively describing the dynamics of short-term interest rates. The Lie algebra of infinitesimal symmetries and the corresponding one-parameter symmetry groups of the equation are obtained. An optimal system of invariant solutions are constructed by a derived optimal system of one-dimensional subalgebras. Because of taking into account a supply response to price rises, this equation provides for a more realistic assumption than the geometric Brownian motion in many investment scenarios.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 1.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical application of Lie symmetry analysis and optimal systems to the partial differential equation governing a classical geometric mean reversion process. However, it is entirely analytical with zero empirical testing, backtesting, or empirical data validation.

Research Flowchart

  flowchart TD
    A[Goal: Analyze Classical Geometric Mean Reversion Process via Lie Symmetry] --> B[Input: Feynman-Kac Formula for Short-Term Interest Rates]
    B --> C[Method: Apply Lie Symmetry Method]
    C --> D[Compute Lie Algebra of Infinitesimal Symmetries]
    D --> E[Construct 1D Subalgebra Optimal System]
    E --> F[Derive Optimal System of Invariant Solutions]
    F --> G[Outcome: Realistic Model Accounting for Price Supply Responses]