Asymptotic inference for skewed stable Ornstein-Uhlenbeck process

TL;DR

This paper develops asymptotic inference methods for a skewed stable Ornstein-Uhlenbeck process driven by a non-Gaussian Lévy process, establishing local asymptotic mixed normality and proposing efficient estimation techniques.

Contribution

It introduces a novel asymptotic analysis for skewed stable OU processes and proposes a simple moment-based initial estimator for the stable Lévy process parameters.

Findings

The parametric family satisfies local asymptotic mixed normality.

Existence of a local maximum of the likelihood that is asymptotically mixed-normal.

Simulation results confirm theoretical properties and compare likelihood methods.

Abstract

We consider the parametric estimation of the Ornstein-Uhlenbeck process driven by a non-Gaussian $\alpha$-stable L\'{e}vy process with the stable index $\alpha>1$ and possibly skewed jumps, based on a discrete-time sample over a fixed period. By employing a suitable non-diagonal normalizing matrix, we present the following: the parametric family satisfies the local asymptotic mixed normality with a non-degenerate Fisher information matrix; there exists a local maximum of the log-likelihood function which is asymptotically mixed-normal; the local maximum is asymptotically efficient in the sense that it has maximal concentration around the true value over symmetric convex Borel subsets. In the proof, we prove the asymptotic equivalence between the genuine likelihood and the much simpler Euler-type quasi-likelihood. Furthermore, we propose a simple moment-based method to estimate the parameters of the driving stable L\'{e}vy process, which serves as an initial estimator for numerical search of the (quasi-)likelihood, reducing the computational burden of the optimization to a large extent. We also present simulation results, which illustrate the theoretical results and highlight the advantages and disadvantages of the genuine and quasi-likelihood approaches.