State-space models with finite dimensional dependence

We consider nonlinear state-space models, where the state variable (ζ_t) is Markov, stationary and features finite dimensional dependence (FDD), i.e. admits a transition function of the type: π(ζ_t|ζ_t-1) = π(ζ_t)a′(ζ_t)b(ζ_t-1), where π(ζ_t) denotes the marginal distribution of ζ_t, with a finite number of cross-effects between the present and past values. We discuss various characterizations of the FDD condition in terms of the predictor space and nonlinear canonical decomposition. The FDD models are shown to admit explicit recursive formulas for filtering and smoothing of the observable process, that arise as an extension of the Kitagawa approach. The filtering and smoothing algorithms are given in the paper.