Real-Time Solution of Linear Least-Squares Estimation Problem with Semi-Degenerate Covariance

On making use of an invariant imbedding, an initial-value solution of the recursive Hopf type integral equations with semi-degenerate asymmetric kernels is presented for the linear least-squares filtering problems with time-variant properties. Furthermore, it is shown that the real-time solution of the optimal estimate is reduced to a Cauchy system for an integral of the stochastic processes weighted with an auxiliary function and also that in the case of displacement kernels the Sobolev resolvent equation is rigorously obtained with the aid of Busbridge´s combined operations method.