Levinson-Durbin-type algorithms for continuous-time autoregressive models and applications (Q1176540)

The random continuous-time autoregressive process described by the stochastic differential equation \[ dx^{(p-1)}(t)+\sum^ p_{j=1}a_ jx^{(p-j)}(t)dt=\sigma dw(t),\qquad t\in R\leqno(1) \] is considered, where \(w(t)\), \(t\in R\) is a standard Wiener process, \(a_ j\), \(j=1,\dots,p\) are constants and \(x^{(i)}(\cdot)\) is the \(i\)th derivative. The equation (1) can be writen as follows \(dX(t)=AX(t)dt+\sigma\ell_ pdw(t)\), where \[ A=\begin{pmatrix} 0&1&0&\dots&0\\ \cdot&\cdot&\cdot&\cdots&\cdot\\ 0&0&0&\ldots&1\\ -a_ p&-a_{p- 1}&a_{p-2}&\dots&-a_ 1\end{pmatrix},\qquad e_ p=\begin{pmatrix} 0\\ \vdots\\ 0\\ 1\end{pmatrix}. \] It is shown, that the connection between \(A\), \(\sigma\) and the covariance \(p\times p\)-matrix \(Q\) is determined by the correlation (2) \(AQ+QA'==\sigma^ 2\ell_ p\ell_ p'\), where prime denotes the transpose. The equation (2) may be written in scalar form as follows (3) \(Q_{i+1,j}+Q_{i,j+1}=0\), \(i,j=0,\dots,p-1\), (4) \(\sum^ p_{j=0}a_ jQ_{p-j,p-k}=0\), \(k=2,\dots,p\), (5) \(\sum^ p_{j=1}a_ jQ_{p-j,p-1}=\sigma^ 2/2\). The recurrence algorithms of the solution of the systems of the algebraic equations (3)-(5) under various assumptions are proposed: either \(a_ j\), \(\sigma^ 2\) are given and it is necessary to find \(Q_{ij}\) or \(Q_{ij}\) are given and it is necessary to find \(a_ j\), \(j=1,\dots,p\).