The Covariance Extension Equation Revisited

In this paper we study the steady state form of a discrete-time matrix Riccati-type equation, connected to the rational covariance extension problem and to the partial stochastic realization problem. This equation, however, is non-standard in that it lacks the usual kind of definiteness properties which underlie the solvability of the standard Riccati equation. Nonetheless, we prove the existence and uniqueness of a positive semidefinite solution. We also show that this equation has the proper geometric attributes to be solvable by homotopy continuation methods, which we illustrate in several examples.