On Markovian extensions of a random processes

The problem on existence of a minimal Markovian process which contains a given random process as a component is considered. Martingale and Markov (in the wide sense) versions are analysed and the existence of the minimal extension as well as its explicit form are established. It is shown, in particular, that the future/past «splitting» subspace of a multivariate stationary process is finite-dimensional if and only if the process has a rational spectral density matrix.