Recursive input-output and state-space solutions for continuous-time linear estimation problems

A general linear least-squares estimation problem is considered. It is shown how the optimal filters for filtering and smoothing can be recursively and efficiently calculated under certain structural assumptions about the covariance functions involved. This structure is related to an index known as the displacement rank, which is a measure of non-Toeplitzness of a covariance kernel. When a state space type structure is added, it is shown how the Chandrasekhar equations for determining the gain of the Kalman-Bucy filter can be derived directly from the covariance function information; thus we are able to imbed this class of state-space problems into a general input-output framework.