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In the paper, we consider nonlinear filtering problems of multiscale systemsin 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 thehomogenized system in the uniform mean square sense. And then based on theconvergence result, in the case of correlated sensor L\'evy noises, thenonlinear filtering of the slow part is shown to approximate that of thehomogenized system in $L^1$ sense. However, in the case of correlated L\'evynoises, we prove that the nonlinear filtering of the slow part converges weaklyto that of the homogenized system.

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