Convergence of nonlinear filterings for multiscale systems with correlated L\'evy noises

TL;DR

This paper studies the convergence of nonlinear filters in multiscale systems with correlated Lévy noises, demonstrating homogenization and approximation results for different noise correlations.

Contribution

It proves the convergence of the slow component to a homogenized system and establishes filtering approximation results under correlated Lévy noises.

Findings

Slow part converges to homogenized system in mean square sense.

Filtering of slow part approximates homogenized filtering in L^1 sense for sensor noises.

Filtering converges weakly for correlated Lévy noises.

Abstract

In the paper, we consider nonlinear filtering problems of multiscale systems in two cases-correlated sensor L\'evy noises and correlated L\'evy noises. First of all, we prove that the slow part of the origin system converges to the homogenized system in the uniform mean square sense. And then based on the convergence result, in the case of correlated sensor L\'evy noises, the nonlinear filtering of the slow part is shown to approximate that of the homogenized system in $L^1$ sense. However, in the case of correlated L\'evy noises, we prove that the nonlinear filtering of the slow part converges weakly to that of the homogenized system.