Stochastic Realization of Gaussian Systems

The weak stochastic realization problem is to determine all stochastic systems whose output equals a considered output process in terms of its finite-dimensional distributions. Such a system is then said to be a stochastic realization of the considered output process. The problem encompasses: (1) an equivalent condition for the existence of a realization, (2) characterizing when such a system is a minimal stochastic realization, and (3) classifying all minimal stochastic realizations. The concepts of stochastic realization theory are the basis of filter theory and of control theory. In this chapter the stochastic realization is presented for stationary Gaussian stochastic processes. Particular concepts discussed include: covariance realization, minimality of a stochastic realization, a classification map, and a canonical form for a stochastic system.