A hardy space approach to the stochastic realization problem

Given a purely nondeterministic mean-square continuous Gaussian stationary stochastic process we consider the problem of characterizing all minimal splitting subspaces X which evolve in time in a Markovian fashion. Let H+/- and H-/+ be the projection of the future of the given process onto the past and the past onto the future respectively. It is shown that the family {X} of minimal Markovian splitting subspaces can be isomorphically described as a partially ordered family of subspaces of the form X ≅ jX* where X* ≅ H-/+ and j ranges over the family of all inner divisors of a fixed inner function j* uniquely defined by H+/-. The procedure is illustrated with an application to a process with a rational spectral density.