Effective Filtering for Multiscale Stochastic Dynamical Systems driven by L\'evy processes

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.