Asymptotic Properties of an Estimator of the Drift Coefficients of Multidimensional Ornstein-Uhlenbeck Processes that are not Necessarily Stable

TL;DR

This paper analyzes the consistency and efficiency of an estimator for the drift matrix of multidimensional Ornstein-Uhlenbeck processes, including unstable, stable, and marginal cases, providing comprehensive asymptotic properties.

Contribution

It extends the understanding of drift estimator properties to all stability regimes of Ornstein-Uhlenbeck processes, including non-stable cases.

Findings

Estimator is consistent across all regimes.

Estimator is asymptotically efficient.

Results cover processes with eigenvalues in all half-planes.

Abstract

In this paper, we investigate the consistency and asymptotic efficiency of an estimator of the drift matrix, $F$, of Ornstein-Uhlenbeck processes that are not necessarily stable. We consider all the cases. (1) The eigenvalues of $F$ are in the right half space (i.e., eigenvalues with positive real parts). In this case the process grows exponentially fast. (2) The eigenvalues of $F$ are on the left half space (i.e., the eigenvalues with negative or zero real parts). The process where all eigenvalues of $F$ have negative real parts is called a stable process and has a unique invariant (i.e., stationary) distribution. In this case the process does not grow. When the eigenvalues of $F$ have zero real parts (i.e., the case of zero eigenvalues and purely imaginary eigenvalues) the process grows polynomially fast. Considering (1) and (2) separately, we first show that an estimator, $\hat{F}$, of $F$ is consistent. We then combine them to present results for the general Ornstein-Uhlenbeck processes. We adopt similar procedure to show the asymptotic efficiency of the estimator.