On the Partial Stochastic Realization Problem*

In this paper we describe a complete parameterization of the solutions to the partial stochastic realization problem in terms of a nonstandard matrix Riccati equation. Our analysis of this Covariance Extension Equation is based on a recent complete parameterization of all strictly positive real solutions to the rational covariance extension problem, answering a conjecture due to Georgiou in the affirmative. We also compute the dimension of partial stochastic realizations in terms of the rank of the unique positive semi-definite solution to the Covariance Extension Equation, yielding some insights into the structure of solutions to the minimal partial stochastic realization problem. By combining this parameterization with some of the classical approaches in partial realization theory, we are able to derive new existence and robustness results concerning the degrees of minimal stochastic partial realizations. As a corollary to these results, we note that, in sharp contrast with the deterministic case, there is no generic value of the degree of a minimal stochastic realization of partial covariance sequences of fixed length.