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2008 Asymptotic properties of an estimator of the drift coefficients of multidimensional Ornstein-Uhlenbeck processes that are not necessarily stable
Gopal K. Basak, Philip Lee
Electron. J. Statist. 2: 1309-1344 (2008). DOI: 10.1214/08-EJS290

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, , 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.

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Gopal K. Basak. Philip Lee. "Asymptotic properties of an estimator of the drift coefficients of multidimensional Ornstein-Uhlenbeck processes that are not necessarily stable." Electron. J. Statist. 2 1309 - 1344, 2008. https://doi.org/10.1214/08-EJS290

Information

Published: 2008
First available in Project Euclid: 22 December 2008

zbMATH: 1320.62192
MathSciNet: MR2471289
Digital Object Identifier: 10.1214/08-EJS290

Subjects:
Primary: 62M05
Secondary: 60F15

Keywords: Asymptotic efficiency , consistency , drift coefficient matrix , estimation , Ornstein-Uhlenbeck processes , Stable process

Rights: Copyright © 2008 The Institute of Mathematical Statistics and the Bernoulli Society

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