Exact filtering and smoothing in short or long memory stochastic switching systems

Let X be a hidden real stochastic chain, R be a discrete finite Markov chain, Y be an observed stochastic chain. In this paper we address the problem of filtering and smoothing in the presence of stochastic switches where the problem is to recover both R and X from Y. In the classical conditionally Gaussian state space models, exact computing with polynomial complexity in the time index is not feasible and different approximations are used. Different alternative models, in which the exact calculations are feasible, have been recently proposed since 2008. The core difference between these models and the classical ones is that the couple (R, Y) is a Markov one in the recent models, while it is not in the classical ones. Another extension deals with the case in which the observed chain Y is not necessarily Markovian conditionally on (X, R) and, in particular, the long-memory distributions can be considered. The aim of this paper is to show that, in the context of these different recent models, it is possible to calculate any moments of the posterior marginal distribution, which makes it feasible to know these distributions with any desired precision.