Asymptotic inference for skewed stable Ornstein-Uhlenbeck process

TL;DR

The paper develops a rigorous local asymptotic theory for parameter estimation in a skewed stable Ornstein–Uhlenbeck process driven by $α$-stable noise with $α>1$, using high-frequency data over a fixed horizon. It shows that a non-diagonal normalization yields local asymptotic mixed normality and efficiency for the maximum likelihood estimator, and that an Euler-type quasi-likelihood provides a tractable yet asymptotically equivalent alternative. A simple moment-based procedure estimates the driving-noise parameters $(α,σ,β)$ to seed drift-parameter optimization, balancing accuracy and computation, and simulations demonstrate the finite-sample behavior and relative merits of genuine versus quasi-likelihood methods. The work also extends to time-scale modeling, showing how a latent time scale can be incorporated and estimated via a two-step approach, with implications for robust inference in non-Gaussian OU-type models.

Abstract

We consider the parametric estimation of the Ornstein-Uhlenbeck process driven by a non-Gaussian $α$-stable Lévy process with the stable index $α>1$ 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 Lévy 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.

Figures (8)

• Figure 1: Comparison between histograms of the normalized MLE and Euler-QMLE with $\alpha=1.0$, $(\lambda,\mu,\sigma,\beta)=(1,2,5,0.5)$.
• Figure 2: Comparison between histograms of the normalized MLE and Euler-QMLE with $\alpha=1.01$, $(\lambda,\mu,\sigma,\beta)=(1,2,5,0.5)$.
• Figure 3: Comparison between histograms of the normalized MLE and Euler-QMLE with $\alpha=1.5$, $(\lambda,\mu,\sigma,\beta)=(1,2,5,0.5)$.
• Figure 4: Comparison between histograms of the normalized MLE and Euler-QMLE with $\alpha=1.8$, $(\lambda,\mu,\sigma,\beta)=(1,2,5,0.5)$.
• Figure 5: Histograms of parameter estimators (Euler-QMLE), $\alpha=1.0$. Each row corresponds to a parameter, and columns correspond to $n=500,1000,2000$.

Theorems & Definitions (35)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Remark 2.4
• Theorem 3.1
• Remark 4.1
• Remark 4.2