Large deviations for interacting particle systems: Applications to non-linear filtering

The non-linear filtering problem consists in computing the conditional distributions of a Markov signal process given its noisy observations. The dynamical structure of such distributions can be modelled by a measure valued dynamical Markov process. Several random particle approximations were recently suggested to approximate recursively in time the so-called non-linear filtering equations. We present an interacting particle system approach and we develop large deviations principles for the empirical measures of the particle systems. We end this paper extending the results to an interacting particle system approach which includes branchings.