Stochastic Realization of Gaussian Systems

Jan H. van Schuppen*

*Corresponding author for this work

Research output: Chapter in Book/Conference proceedings/Edited volumeChapterScientific


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.

Original languageEnglish
Title of host publicationControl and System Theory of Discrete-Time Stochastic Systems
EditorsJ.H. van Schuppen
Number of pages48
ISBN (Electronic)978-3-030-66952-2-6
Publication statusPublished - 2021

Publication series

NameCommunications and Control Engineering
ISSN (Print)0178-5354
ISSN (Electronic)2197-7119


  • Gaussian systems
  • Stochastic realization


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