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

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.