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

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.