On the Convergence Analysis of Yau-Yau Nonlinear Filtering Algorithm: from a Probabilistic Perspective

TL;DR

This paper provides a new probabilistic convergence analysis of the Yau-Yau nonlinear filtering algorithm, demonstrating its accuracy in approximating key statistics of the conditional distribution under broad conditions.

Contribution

It generalizes previous results and introduces a probabilistic approach to analyze the convergence of the Yau-Yau algorithm, focusing on expectation of approximation errors.

Findings

Yau-Yau algorithm accurately approximates conditional mean and covariance.

The probabilistic convergence analysis applies under very general nonlinear systems.

Provides theoretical guidance for practical implementation of the algorithm.

Abstract

At the beginning of this century, a real time solution of the nonlinear filtering problem without memory was proposed in [1, 2] by the third author and his collaborator, and it is later on referred to as Yau-Yau algorithm. During the last two decades, a great many nonlinear filtering algorithms have been put forward and studied based on this framework. In this paper, we will generalize the results in the original works and conduct a novel convergence analysis of Yau-Yau algorithm from a probabilistic perspective. Instead of considering a particular trajectory, we estimate the expectation of the approximation error, and show that commonly-used statistics of the conditional distribution (such as conditional mean and covariance matrix) can be accurately approximated with arbitrary precision by Yau-Yau algorithm, for general nonlinear filtering systems with very liberal assumptions. This novel probabilistic version of convergence analysis is more compatible with the development of modern stochastic control theory, and will provide a more valuable theoretical guidance for practical implementations of Yau-Yau algorithm.