Convergence of nonlinear filtering for multiscale systems with correlated Lévy noises

Abstract

The work concerns nonlinear filtering problems of multiscale systems with Lévy noises in two cases-correlated noises and correlated sensor noises. First of all, we prove that the slow part of the origin system converges to the average system in the uniform mean square sense. Then based on the convergence result, in two cases of correlated Lévy noises and correlated Gaussian noises, we prove that the nonlinear filtering of the slow part converges to that of the average system in the weak sense and the [Formula: see text] sense, respectively. Finally, in the case of correlated sensor Gaussian noises, the nonlinear filtering for the slow part is shown to approximate that of the average system in the [Formula: see text] sense.