Continuous Finite-Dimensional Locally Optimal Filtering of Jump Diffusions

This paper considers continuous-time state estimation for a dynamic system with Gaussian and Poisson white noise. We design a finite-dimensional filter with a lower computational cost than the extended Kalman-Bucy filter: its order is the dimension of the estimated part of the system state vector. The nonlinear structure of the filter is selected using the mean squared optimal unbiased estimation for each infinitesimal period of time. We develop an algorithm to find the structural functions of the filter that is based on the Kolmogorov-Feller equation for the density function. A numerical method to calculate them a priori by sequential Monte Carlo trials is presented, which requires the histograms of the desired functions. Due to its cumbersome form, some numerical and analytical approximations of the suggested filter are also given. They structurally coincide with the corresponding nonlinear extensions of the Kalman-Bucy filter but have a considerably smaller order and calculable parameters.