How to verify that a given process is a L\'evy-Driven Ornstein-Uhlenbeck Process

TL;DR

This paper presents a step-by-step methodology for verifying whether a discrete-time observed process is a Lévý-driven Ornstein-Uhlenbeck process, including parameter estimation, driving process approximation, and hypothesis testing, with applications to financial data.

Contribution

It introduces a novel, practical approach for testing the Lévý-driven Ornstein-Uhlenbeck model assumptions and class membership from discrete observations.

Findings

Method successfully distinguishes Lévý-driven OU processes from other models.

Simulation results demonstrate high accuracy of the proposed tests.

Application to economic data illustrates real-world utility.

Abstract

Assuming that a L\'evy-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 L\'evy-driven. We then show how to test the hypothesis that the CAR(1) process belongs to a specified class of L\'evy 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.