Filtering for a signal given by a linear stochastic retarded differential equation (Q1364828)

This paper is concerned with finding a recursive relation for the estimate \(x(t)= E\{x(t)/y(u)\), \(0\leq u\leq t\}\) of \(x(t)\) in a Kalman-Bucy type problem, where \(y(t)\) is the observable process and \(x(t)\) is the ``hidden'' signal process. The signal \(x(t)\) is found from a linear retarded stochastic differential equation involving the section \(x_t(s)= x(t+s)\). The second equation of the considered filtering problem, giving linearly the observable process depends on \(x(t)\) through its section, too. The work emphasizes the non-discrete delays case and modifies the innovations approach for the Kalman-Bucy filtering problem using facts on retarded equations.