In this paper we analyze the structure of the class of discrete-time linear stochastic systems in terms of the geometric theory of stochastic realization. We discuss the role of invariant directions, zeros of spectral factors, and output-induced subspaces in determining the systems-theoretical properties of the stochastic systems. A prototype interpolation problem for recovering lost state information is discussed, and it is shown how it can be solved via Kalman filtering recursions tying together the state processes of a family of totally ordered splitting subspaces.