TOWARD A SOLUTION OF THE MINIMAL PARTIAL STOCHASTIC-REALIZATION PROBLEM

In this Note we describe a complete parametrization of the solutions to the partial stochastic realization problem in terms of a nonstandard matrix Riccati equation, which is analyzed using topological methods. Our analysis of this Covariance Extension 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 semi-definite solution to the Covariance Extension Equation, yielding some preliminary insights into the structure of solutions to the minimal partial stochastic realization problem.