Estimation and Application of the Convergence Bounds for Nonlinear Markov Chains

TL;DR

This paper introduces a novel method to analyze the ergodicity and estimate convergence bounds of nonlinear Markov chains, improving precision and enabling volatility estimation through coupling techniques and wavelet analysis.

Contribution

It proposes a new approach using Coupling Markov chains to accurately estimate convergence bounds and introduces TV Volatility for financial data analysis.

Findings

Convergence bounds are more precise than existing results.

TV Volatility effectively reflects changes in financial returns.

Method successfully applied to securities TSLA and AMC.

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.