Reciprocal covariance solutions of some matrix differential equations (Q2640993)

Let \(\{\) X(\(\tau\)), \(a\leq \tau \leq b\}\) be a random process. Assume that (*) \(a\leq c<s<t<d\leq b\). Let \({\mathcal E}(s,t)\) be the \(\sigma\)-field generated by \(\{X(r),\quad r\in (c,d)\setminus (s,t)\}\) and \({\mathcal F}(s,t)\) be the \(\sigma\)-field generated by \(\{X(r),\quad r\in (s,t)\}.\) The process X(\(\tau\)) is called (c,d)-reciprocal, if for each s,t satisfying (*), \({\mathcal E}(s,t)\) and \({\mathcal F}(s,t)\) are conditionally independent given X(s) and X(t). It is shown that the covariance matrix function of a multivariate reciprocal stationary Gaussian process with continuous parameters satisfies two related matrix differential equations. The authors solve also the inverse problem, under which conditions two appropriately related matrix differential equations have a common solution that is the covariance matrix function of a reciprocal stationary Gaussian process.