Realization Theory for Multivariate Stationary Gaussian Processes

This paper collects in one place a comprehensive theory of stochastic realization for continuous time stationary Gaussian vector processes which in various pieces has appeared in a number of our earlier papers. It begins with an abstract state space theory, based on the concept of splitting subspace. These results are then carried over to the spectral domain and described in terms of Hardy functions. Finally, differential-equations type stochastic realizations are constructed. The theory is coordinate-free, and it accommodates infinite-dimensional representations, minimality and other systems-theoretical concepts being defined by subspace inclusion rather than by dimension. We have strived for conceptual completeness rather than generality, and the same framework can be used for other types of stochastic realization problems.