Recursive filtering and smoothing for Gaussian reciprocal processes with continuous indices

In this paper, we study continuous index Gaussian reciprocal processes (Grp´s) (or two-point boundary valued processes) with Dirichlet boundary conditions. Our main contributions are 1) deriving first order white noise driven representations from the second order correlated noise driven representations of Grp´s given by Krener, Frezza, and Levy; 2) deriving Kalman-Bucy like recursive filtering equations for Grp´s with continuous indices; and 3) deriving recursive smoothing equations for Grp´s with continuous indices.