On the behaviour of the Kalman-Bucy estimate if the involved Wiener-Lévy processes are approximated by physically realizable processes

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It is noteworthy that the possibility of computing Kalman-Bucy estimates depends entirely on the mathematical properties of the Wiener-Lévy processes figuring in the observations, whereas no observation device ever could generate processes of that kind. However, there are physically realizable processes arbitrarily close to Wiener-Lévy processes. In this paper the consequences are investigated if the Wiener-Lévy processes in the Kalman-Bucy filter are replaced by realizable approximations. The effect is not simply that of perturbing some values in certain formulae, since the whole computation scheme of Kalman and Bucy breaks down. It is shown that estimates of Kalman-Bucy type, based on a finite number of observations, are stable with respect to the above sketched operations. A more or less controversial result is obtained if the number of observations is infinite.