Hermite polynomials expansions for discrete-time nonlinear filtering

Abstract

A finite-dimensional approximation to general discrete-time non linear filtering problems is provided. It consists in a direct approximation to the recursive Bayes formula, based on a Hermite polynomials expansion of the transition density of the signal process. Tha approximation is in the sense of convergence, in a suitable weighted norm, to the conditional density of a signal process given the observations. The choise of the norm is in turn made so as to guarantee also the convergence of the conditional moments as well as to allow the evaluation of an upper bound for the approximation error.