Finite dimensional filters for nonlinear stochastic difference equations with multiplicative noises

We consider the filtering problem for partially observable stochastic processes {Xn,Yn}n∈N, solutions to systems of stochastic difference equations. In the first part of the paper we shall present a simple constructive method to obtain finite dimensional filters in discrete time. Then, applying some well-known results, mainly on the product of independent positive random variables, we shall present new finite dimensional filters and interpret some known results in a more general setting.