Authors’ Abstract: We describe a complete parametrization of the solution to the partial stochastic realization problem in terms of a nonstandard matrix Riccati equation. Our analysis of this covariance equation (CEE) is based on a recent complete parametrization 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 semidefinite solution to the CEE, yielding some insights into the structure of solutions to the minimal partial stochastic realization problem. By combining this parametrization 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.”
Note of the reviewer: This paper is highly readable, gives motivation, explains very clearly the issues at stake and may serve as well as a nice introduction to the partial stochastic realization problem.