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

TL;DR

This paper investigates how multiscale stochastic systems driven by Lévy processes can be approximated by lower-dimensional systems using invariant manifolds, and explores the limitations of such reductions.

Contribution

It proves the approximation of multiscale Lévy-driven systems by low-dimensional systems and analyzes the effectiveness of nonlinear filtering in this context.

Findings

Multiscale systems can be approximated by low-dimensional invariant manifolds.

Nonlinear filtering of multiscale systems approximates that of reduced systems.

Reduction at zero scale parameter does not approximate the original multiscale systems.

Abstract

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.