Universal Nonlinear Filtering Using Feynman Path Integrals II: The Continuous-Continuous Model with Additive Noise

TL;DR

This paper develops a Feynman path integral approach to the continuous-continuous filtering problem with additive noise, providing a new formula for the conditional density and an improved algorithm for practical implementation.

Contribution

It introduces a novel Feynman path integral formulation for the filtering problem with additive noise, leading to the Yau algorithm and simplified computational methods.

Findings

Yau algorithm outperforms existing algorithms

Path integral formula reveals underlying physics

Reduction of PDE solution to function computation and integration

Abstract

In this paper, the Feynman path integral formulation of the continuous-continuous filtering problem, a fundamental problem of applied science, is investigated for the case when the noise in the signal and measurement model is additive. It is shown that it leads to an independent and self-contained analysis and solution of the problem. A consequence of this analysis is Feynman path integral formula for the conditional probability density that manifests the underlying physics of the problem. A corollary of the path integral formula is the Yau algorithm that has been shown to be superior to all other known algorithms. The Feynman path integral formulation is shown to lead to practical and implementable algorithms. In particular, the solution of the Yau PDE is reduced to one of function computation and integration.