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). 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.