  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.
