Stochastic realization of Gaussian processes

A Gaussian stochastic process (y <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</inf> ) with known covariance kernel is given: we investigate the generation of (y <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</inf> ) by means of Markovian schemes of the type dx <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</inf> = F(t)x <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</inf> dt + dw <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</inf> y <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</inf> = H(t)x <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</inf> . Such a generation of (y <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</inf> ) 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 (y <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</inf> ) 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 (y <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</inf> ). The corresponding realization has been referred to as the "innovation representation" of (y <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</inf> ).