A Gaussian-generalized inverse Gaussian finite-dimensional filter

We consider the filtering problem for a partially observable stochastic pro cess {X-n, Z(n), Y-n}(n is an element of N), solution to a nonlinear system of stochastic difference equations, which provides a stochastic modellizat ion for both the mean and the variance of the Gaussian observation distribu tion. The noises in the equations are given by two sequences of independent Gaussian random variables and a sequence of independent gamma random varia bles. We are able to prove that there exists a finite-dimensional filter sy stem for this model, since, for each n, the conditional distribution of (X- n,Z(n)) given (Y-0,..., Y-n) is that of a suitable bivariate Gaussian-gener alized inverse Gaussian random variable. (C) 1999 Elsevier Science B.V. All rights reserved.