Constructing Non-linear Gaussian Time Series by Means of a Simplified State Space Representation

State space models provide a useful stochastic description for dynamic phenomena, based on unobserved or latent variables. When the model rests on linear and Gaussian assumptions there exists a well-known iterative procedure, called the Kalman filter, which gives analytic updating recursion for the filtering, the prediction and the smoothing distributions. However, this is rare and a state space model does not usually admit such a filter. For this reason, instead of looking for analytic solutions, a number of papers aim to define alternative procedures, giving numerical or approximate solutions. This paper concerns a particular class of models based on the assumption that the mixed process, obtained by alternating states and observations, is a Markov process. The main features of this class of models, proposed for stochastic volatility description by Barndorff-Nielsen (1997), are emphasized. In this framework, some new non-linear Gaussian state space models, computationally tractable and of potential interest for applications, may be defined.