On the Stochastic Realization Problem

TLDR

The paper seeks to characterize all minimal wide‑sense Markov realizations of a mean‑square continuous stochastic vector process with stationary increments and a rational, nonsingular spectral density. An algorithm is introduced that classifies these realizations into internal and external families, orders them by state covariance, and provides a constructive method for generating external realizations on the same probability space. The study shows that only internal realizations can be inferred from the output process, while external realizations require additional randomness, and it offers a complete characterization of both sets and demonstrates that the state of any internal realization can be expressed via two steady‑state Kalman–Busy filters running forward and backward in time.

Abstract

Given a mean square continuous stochastic vector process y with stationary increments and a rational spectral density $\Phi $ such that $\Phi (\infty )$ is finite and nonsingular, consider the problem of finding all minimal (wide sense) Markov representations (stochastic realizations) of y. All such realizations are characterized and classified with respect to deterministic as well as probabilistic properties. It is shown that only certain realizations (internal stochastic realizations) can be determined from the given output process y. All others (external stochastic realizations) require that the probability space be extended with an exogeneous random component. A complete characterization of the sets of internal and external stochastic realizations is provided. It is shown that the state process of any internal stochastic realization can be expressed in terms of two steady-state Kalman–Busy filters, one evolving forward in time over the infinite past and one backward over the infinite future. An algorithm is presented which generates families of external realizations defined on the same probability space and totally ordered with respect to state covariances.