Dimensional reduction in nonlinear filtering: A homogenization approach

TL;DR

This paper introduces a homogenized filtering method for multiscale signals that reduces system dimension and proves convergence of the nonlinear filter to this simplified version with a specific rate, using advanced mathematical tools.

Contribution

The paper presents a novel homogenization approach for nonlinear filtering in multiscale systems, including convergence proofs and correction term representations.

Findings

Nonlinear filter converges to homogenized filter at rate √ε

Homogenized filter effectively reduces system dimension

Asymptotic expansion and BDSDEs are used for correction terms

Abstract

We propose a homogenized filter for multiscale signals, which allows us to reduce the dimension of the system. We prove that the nonlinear filter converges to our homogenized filter with rate $\sqrt{\varepsilon}$. This is achieved by a suitable asymptotic expansion of the dual of the Zakai equation, and by probabilistically representing the correction terms with the help of BDSDEs.