Asymptotic likelihood based inference for co-integrated homogeneous Gaussian diffusions (Q2771550)

The data \((X_s)_{0\leq s\leq t}\) is one trajectory of a solution of the Gaussian diffusion equation \(dX_t=(\mu+\Pi X_t)dt +\Sigma dW_t\), where \(\Pi\) is a \(p\times p\) matrix, \(\mu\) is a \(p\)-dimensional vector, \(\Sigma\) is \(p\times d\), \(d\leq p\) such that \(\Omega=\Sigma\Sigma'\) is nonsingular, and \(W\) is a \(d\)-dimensional Brownian motion. The authors consider tests for the hypothesis \(H_r\): \(\text{rank}(\Pi)\leq r\) and \(\mu\) lies in the column space of \(\Pi\) (the first condition corresponds to the co-integration of \(X\) entries, whence the second one is used to avoid a linear trend in \(X\)). The authors describe maximum likelihood estimators for the parameters under \(H_r\) and the likelihood ratio test for \(H_r\) against \(H_p\). Asymptotic expansions for the estimators are derived.