Continuous-discrete filtering using the Dirac-Feynman algorithm

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 presented for the case the signal model noise is additive. The simplest approximation of the path integral formula leads to the Dirac-Feynman algorithm for continuous-discrete filtering. The practical utility of the proposed Dirac-Feynman algorithm is demonstrated in a fully nonlinear filtering problem where it is shown that it leads to highly accurate and reliable results. The Feynman path integral approach to nonlinear filtering has several advantages, especially in terms of ease in practical implementation.