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

TL;DR

This paper derives a universal path integral formula for solving the Fokker-Planck-Kolmogorov forward equation in continuous-discrete filtering problems with additive noise, enabling flexible initial distributions and practical applications.

Contribution

It introduces a universal path integral solution for the FPKfe in continuous-discrete filtering with additive noise, applicable to arbitrary initial distributions.

Findings

Derived and verified the path integral formula for the FPKfe.

Demonstrated practical utility through examples.

Applicable to time-dependent state models with additive noise.

Abstract

The continuous-discrete filtering problem requires the solution of a partial differential equation known as the Fokker-Planck-Kolmogorov forward equation (FPKfe). In this paper, the path integral formula for the fundamental solution of the FPKfe is derived and verified for the general additive noise case (i.e., explicitly time-dependent state model and with state-independent rectangular diffusion vielbein). The solution is universal in the sense that the initial distribution may be arbitrary. The practical utility is demonstrated via some examples.