Paper: arXiv 2212.05304

Authors: Kaichen Xu

Abstract

Nonlinear Markov Chains (nMC) are regarded as the original (linear) Markov Chains with nonlinear small perturbations. It fits real-world data better, but its associated properties are difficult to describe. A new approach is proposed to analyze the ergodicity and even estimate the convergence bounds of nMC, which is more precise than existing results. In the new method, Coupling Markov about homogeneous Markov chains is applied to reconstitute the relationship between distribution at any times and the limiting distribution. The convergence bounds can be provided by the transition probability matrix of Coupling Markov. Moreover, a new volatility called TV Volatility can be calculated through the convergence bounds, wavelet analysis and Gaussian HMM. It’s tested to estimate the volatility of two securities (TSLA and AMC). The results show TV Volatility can reflect the magnitude of the change of square returns in a period wonderfully.

Complexity vs Empirical Score

  • Math Complexity: 7.5/10
  • Empirical Rigor: 4.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper introduces a rigorous theoretical framework using Coupling Markov chains to derive tighter convergence bounds for nonlinear Markov chains, demonstrating strong mathematical novelty. However, the empirical evaluation is limited to a brief application on just two individual stocks (TSLA and AMC) without comprehensive out-of-sample backtesting or robustness checks.

Research Flowchart

  flowchart TD
    A[Research Goal: Analyze nMC Ergodicity & Estimate TV Volatility] --> B[Data Inputs: TSLA & AMC Stock Returns]
    B --> C[Coupling Markov Approach: Reconstitute Distribution Relationship]
    C --> D[Estimate Precise nMC Convergence Bounds]
    D --> E[Compute TV Volatility via Wavelet Analysis & Gaussian HMM]
    E --> F[Outcome: Accurately Reflects Magnitude of Return Changes]