Stability and particle approximations in nonlinear filtering applied to tracking.

Summary The problem of nonlinear filtering consists in computing in an approximate way the conditional law of a Markovian process called signal, indirectly related to an observational process of which a realization is known. The aim of this thesis is to propose new particle algorithms for the solution of the nonlinear filtering problem. The first idea developed here underlines the strong link between the stability properties of the optimal filter and the long time behavior of some approach filters. We study the sensitivity of the optimal filter to different types of local perturbations intervening in its evolution at each time step. The key role of the Hilbert projective metric is highlighted, which allows the time-uniform control of the global error induced in the perturbed filter provided that the signal verifies certain ergodicity conditions. In particular, these results show that under these same ergodicity conditions, the two particle filters initially proposed in the literature (weighted monte carlo filter and particle filter with interaction) converge uniformly in time, towards the optimal filter. However, a more general analysis of the classical particle methods shows their weakness especially in the case of low noise systems and leads us to propose a new type of particle filters using a finer local perturbation. Regularization is the key step of this perturbation. It is based on an extension of the theory of density estimation by kernels and allows to replace the discrete approximation provided by the classical particle filters in a smooth approximation. The two types of resulting particle filters called pre-regularized and post-regularized filters are analyzed. They are then applied in simulations to different tracking problems.