A Gaussian stochastic process (yt) with known covariance kernel is given: we investigate the generation of (yt) by means of Markovian schemes of the type dxt= F(t)xtdt + dwtyt= H(t)xt. Such a generation of (yt) as the "output of a linear dynamical system driven by white noise" is possible under certain finiteness conditions. In fact, this was shown by Kalman in 1965. We emphasize the probabilistic aspects and obtain an intrinsic characterization of the state of the process as the state of an externally described stochastic I/O map. Realizations of (yt) can be constructed with respect to any increasing family of ω-fields; in particular, when the family of ω-fields is induced by the process itself, the driving white noise reduces to the innovation process of (yt). The corresponding realization has been referred to as the "innovation representation" of (yt).