Mean-square filtering problem for stochastic polynomial systems with Gaussian and Poisson noises

This paper presents the mean-square finite-dimensional filter for polynomial system states confused with both, Gaussian and Poisson, white noises over linear observations. Designing the mean-square filter for polynomial systems with white Gaussian and Poisson noises enables one to address the mean-square filtering problems for nonlinear system states confused not only with Gaussian white noises but arbitrary strictly defined white noises being weak mean-square derivatives of martingales. A procedure is established for designing the optimal filtering equations for system states described by polynomial equations of an arbitrary finite degree. An explicit closed form of the designed filter is obtained in case of a third-order polynomial. Performance of the designed optimal filter is verified for a third degree polynomial state.