A Gaussian-generalized inverse Gaussian finite-dimensional filter

We consider the filtering problem for a partially observable stochastic process , solution to a nonlinear system of stochastic difference equations, which provides a stochastic modellization for both the mean and the variance of the Gaussian observation distribution. The noises in the equations are given by two sequences of independent Gaussian random variables and a sequence of independent gamma random variables. We are able to prove that there exists a finite-dimensional filter system for this model, since, for each n, the conditional distribution of (Xn,Zn) given (Y0,...,Yn) is that of a suitable bivariate Gaussian-generalized inverse Gaussian random variable.