A note on the integral-representation of the Kalman-Bucy estimate

AbstractThe Wiener-Hopf equations of the Kalman-Bucy estimate can be solved, owing to the assumption that the covariance matrix of the observation noise is invertible. It will be shown that this property is already needed from the very outset, where the estimate is represented as a stochastic integral. This representation will be characterized by means of a generalization of a theorem of Karhunen. Then it can be shown that the usual formal derivations, leading to the filter equations, are legitimate