How to verify that a given process is a Lévy-Driven Ornstein-Uhlenbeck Process

ArXiv ID: 2501.03434 “View on arXiv”

Authors: Unknown

Abstract

Assuming that a Lévy-Driven Ornstein-Uhlenbeck (or CAR(1)) processes is observed at discrete times $0$, $h$, $2h$,$\cdots$ $[“T/h”]h$. We introduce a step-by-step methodological approach on how a person would verify the model assumptions. The methodology involves estimating the model parameters and approximating the driving process. We demonstrate how to use the increments of the approximated driving process, along with the estimated parameters, to test the assumptions that the CAR(1) process is Lévy-driven. We then show how to test the hypothesis that the CAR(1) process belongs to a specified class of Lévy processes. The performance of the tests is illustrated through multiple simulations. Finally, we demonstrate how to apply the methodology step-by-step to a variety of economic and financial data examples.

Keywords: Lévy-Driven Ornstein-Uhlenbeck, CAR(1) Process, Hypothesis Testing, Stochastic Processes, Parameter Estimation, Econometrics / General Financial Data

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 7.0/10
  • Quadrant: Holy Grail
  • Why: The paper is densely packed with advanced stochastic process theory, Lévy process definitions, and asymptotic statistical derivations, pushing its math score high. It demonstrates strong empirical rigor by providing a step-by-step methodology, evaluating test performance via simulations, and applying the method to real financial datasets (S&P 500, Euro/USD).
  flowchart TD
    A["Research Goal: Verify Lévy-Driven<br>Ornstein-Uhlenbeck Process Assumptions"] --> B{"Data Input:<br>Discrete Observations"}
    B --> C["Step 1: Parameter Estimation<br>and Driving Process Approximation"]
    C --> D["Step 2: Extract Increments<br>from Approximated Driving Process"]
    D --> E["Step 3: Hypothesis Testing<br>Using Increments & Parameters"]
    E --> F{"Outcomes:<br>Test Statistics & P-Values"}
    F --> G["Conclusion:<br>Accept/Reject Lévy Assumption<br>and Specific Lévy Class"]