Weak approximation of nonlinear filtering for multiscale McKean-Vlasov stochastic systems

TL;DR

This paper studies the weak approximation of nonlinear filtering in multiscale McKean-Vlasov stochastic systems, demonstrating convergence of the slow component's filter to that of an averaged system using Poisson equations.

Contribution

It introduces a novel approach to approximate nonlinear filters for multiscale systems via weak convergence analysis using Poisson equations.

Findings

Weak convergence of the slow component's filter to the averaged system's filter.

Application of Poisson equations to establish approximation results.

Theoretical validation of filtering approximation in multiscale stochastic systems.

Abstract

The work concerns the nonlinear filtering problem for a class of multiscale McKean-Vlasov stochastic systems. First of all, by a Poisson equation we prove that the solution of the slow part for a multiscale system weakly converges to the solution of the average equation. Then we define nonlinear filtering of the origin multiscale system and the average equation, and again through the same Poisson equation show the weak approximation between nonlinear filtering of the slow part for the origin multiscale system and that of the average equation.