Asymptotic properties of an estimator of the drift coefficients of multidimensional Ornstein-Uhlenbeck processes that are not necessarily stable

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 eigen-values 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, (F) over cap, 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.