Finite dimensional filters for nonlinear stochastic difference equations with multiplicative noises

We consider the filtering problem for partially observable stochastic processes 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.