Paper: arXiv 2409.04496

Authors: Mohamed Ben Alaya, Martin Friesen, Jonas Kremer

Abstract

We study the Volterra Volterra Cox-Ingersoll-Ross process on $\mathbb{R}_+$ and its stationary version. Based on a fine asymptotic analysis of the corresponding Volterra Riccati equation combined with the affine transformation formula, we first show that the finite-dimensional distributions of this process are asymptotically independent. Afterwards, we prove a law-of-large numbers in $L^p$(Ω)$ with $p \geq 2$ and show that the stationary process is ergodic. As an application, we prove the consistency of the method of moments and study the maximum-likelihood estimation for continuous and discrete high-frequency observations.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical treatment of ergodicity and the law of large numbers for the Volterra Cox-Ingersoll-Ross (VCIR) process, solving key open theoretical problems for non-Markovian stochastic Volterra processes. While the work is strong theoretically and includes applications to parameter estimation (MLE and method of moments), it lacks empirical backtesting or real market data validation.

Research Flowchart

  flowchart TD
    A[Research Goal: Ergodicity & Parameter Estimation for Volterra CIR Process] --> B[Input: Volterra CIR Model & Affine Transformation Formula]
    B --> C[Methodology: Asymptotic Analysis of Volterra Riccati Equation]
    C --> D[Prove Asymptotic Independence of Finite-Dimensional Distributions]
    D --> E[Establish Lp Law of Large Numbers & Ergodicity for Stationary Process]
    E --> F[Key Outcome: Consistency of Method of Moments & Maximum-Likelihood Estimation]