A New Proof for the Linear Filtering and Smoothing Equations, and Asymptotic Expansion of Nonlinear Filtering

TL;DR

This paper introduces an asymptotic expansion method for nonlinear filtering that improves computational efficiency and accuracy, unifies classical linear filtering equations, and offers a systematic way to approximate filtering distributions.

Contribution

It presents a novel asymptotic expansion approach based on a small system noise parameter, unifying linear and nonlinear filtering equations through explicit formulas.

Findings

Derives a power series expansion for nonlinear filtering distributions.

Shows classical linear filters can be derived from the new framework.

Provides a computational method using differential equations for coefficients.

Abstract

In this paper, we propose a new asymptotic expansion approach for nonlinear filtering based on a small parameter in the system noise. This method expresses the filtering distribution as a power series in the noise level, where the coefficients can be computed by solving a system of ordinary differential equations. As a result, it addresses the trade-off between computational efficiency and accuracy inherent in existing methods such as Gaussian approximations and particle filters. In the course of our derivation, we also show that classical linear filtering and smoothing equations, namely Kalman-Bucy filter and Rauch-Tung-Striebel smoother, can be obtained in a unified and transparent manner from an explicit formula for the conditional distribution of the hidden path.