Abstract
The existence and the explicit form of the minimal Markov process which contains as a component a given stationary process are established. It is shown in particular that the future/past splitting subspace of the multivariate stationary process is finite-dimensional if and only if the process has a rational spectral densities matrix. The property of being stochastically continuous is obtained as the condition for continuation of the sigma-algebras associated with a process with independent increments which usually represents a stochastic disturbance of the system considered. This property gives us the left-side continuation of the sigma-algebras in the case of the arbitrary process in a metric space.
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