On the structure of minimal splitting subspaces in stochastic realization theory

The problem of determining all (internal) Markovian representations (realizations) for a Gaussian stochastic process with stationary increments and rational spectral density is resolved. A complete characterization of all minimal splitting subspaces (state spaces) is presented, and it is shown that these are completely determined by the numerator polynomial of the spectral density and the degree of the denominator polynomial. This provides a coordinate-free solution of the stochastic realization problem; any state space basis forms a Markovian state vector process. If a differential equation for the state process is required, the denominator polynomial enters the analysis. A complete characterization of all such realizations is given.