  The work is about multiscale stochastic dynamical systems driven by L\'evy
processes. First, we prove that these systems can approximate low-dimensional
systems on random invariant manifolds. Second, we establish that nonlinear
filterings of multiscale stochastic dynamical systems also approximate that of
reduced low-dimensional systems. Finally, we investigate the reduction for
$\e=0$ and obtain that these reduced systems does not approximate these
multiscale stochastic dynamical systems.
