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Finite dimensional filters for random parameter AR models

In this paper exact finite dimensional filters are derived for a class of doubly stochastic autoregressive models. The parameters of the doubly stochastic autoregressive process vary according to a nonlinear function of a Gauss-Markov process. We develop a difference equation for the evolution of an unnormalized conditional density related to the state of the doubly stochastic autoregressive process. We then give a characterization of the general solution followed by examples for which the state of the filter is determined by a finite number of sufficient statistics. These new finite dimensional filters are built upon the discrete-time Kalman filter.