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Exact Filtering and Smoothing in Markov Switching Systems Hidden with Gaussian Long Memory Noise

Let X be a hidden real stochastic process, R be a discrete finite Markov chain, Y be an observed chain. The problem of filtering and smoothing is the problem of recovering both R and X from Y. In the classical models the exact computing with linear - or even polynomial - complexity in time index is not feasible and different approximations are used. Different alternative models, in which the exact calculations are feasible, have been recently proposed (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 all the classical ones. Here we propose a further extension of these models. The core point of the extension is the fact that the observed chain Y is not necessarily Markovian conditionally on (X, R) and, in particular, the long-memory distributions can be considered. We show that both filtering and smoothing are computable with complexity polynomial in the number of observations in the new model.