Unimodal maps perturbed by heteroscedastic noise: an application to a financial systems

Paper: arXiv 2305.13475

Authors: F. Lillo, G. Livieri, S. Marmi, A. Solomko, S. Vaienti

Abstract

We investigate and prove the mathematical properties of a general class of one-dimensional unimodal smooth maps perturbed with a heteroscedastic noise. Specifically, we investigate the stability of the associated Markov chain, show the weak convergence of the unique stationary measure to the invariant measure of the map, and show that the average Lyapunov exponent depends continuously on the Markov chain parameters. Representing the Markov chain in terms of random transformation enables us to state and prove the Central Limit Theorem, the large deviation principle, and the Berry-Essèen inequality. We perform a multifractal analysis for the invariant and the stationary measures, and we prove Gumbel’s law for the Markov chain with an extreme index equal to 1. In addition, we present an example linked to the financial concept of systemic risk and leverage cycle, and we use the model to investigate the finite sample properties of our asymptotic results.

Complexity vs Empirical Score

• Math Complexity: 9.2/10
• Empirical Rigor: 4.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical framework establishing Markov chain properties, Lyapunov exponents, and extreme value theory for unimodal maps with heteroscedastic noise. While it applies this heavy mathematical apparatus to systemic risk and leverage cycles, the empirical validation relies primarily on finite-sample numerical simulations rather than extensive real-world financial datasets.

Research Flowchart

  flowchart TD
    A[Research Goal: Analyze 1D Unimodal Maps<br/>Perturbed by Heteroscedastic Noise] --> B[Model Formulation:<br/>Markov Chain as Random Transformation]
    B --> C[Inputs: Smooth Unimodal Map &<br/>State-Dependent Heteroscedastic Noise]
    C --> D[Mathematical & Asymptotic Analysis:<br/>Stability, Lyapunov Exponents, LDP, CLT]
    C --> E[Multifractal & Extreme Value Analysis:<br/>Gumbel's Law & Fractal Dimensions]
    D --> F[Financial Application:<br/>Systemic Risk & Leverage Cycle Simulation]
    E --> F
    F --> G[Key Outcomes: Stationary Measure Convergence,<br/>Continuous Exponents, EVT Proof & Finite Sample Validation]