Linear Kalman-Bucy ﬁlter with an autoregressive signal and noise
In the Kalman-Bucy ﬁlter problem the observed process consists of a sum of a signal and of a noise. The ﬁlter begins simultaneously with the beginning of observation, and it is necessary to estimate a signal. As a rule, this problem is studied both for scalar and for vector Markovian processes. In this paper, the scalar linear problem is considered, but a signal and a noise are independent stationary autoregressive processes with orders larger than unit. The recurrent equations for the ﬁlter process, for its error, and for its conditioncorrelations are obtained. These recurrent equations contain the previous estimates and some last observed data. The optimal deﬁnition of initial data is proposed. The algebraic equations for the limit values of the ﬁlter error and for the limit cross-correlations are delivered. The roots of these equations lead to a criterion of the ﬁlter process convergence. Some examples at that the ﬁlter process converges or not converges are given. The Monte-Carlo method is used for a control of theoretical formulas for the ﬁlter and its error.