Recursive nonlinear estimation of random parameter AR models with Poisson observations

We derive exact filters for the state of a doubly stochastic AR process with parameters which vary according to a nonlinear function of a Gauss-Markov process. The observations consist of a discrete time Poisson process with rate a positive function of the Gauss-Markov process. The dimension of the sufficient statistic increases linearly with the number of observed events.