Statistical algorithms of optimal filtering problem for nonlinear jump-diffusion models

Abstract

 New statistical algorithm and its modifications for solving the optimal nonlinear filtering problem are described. It is assumed that the observation object and measurement system are described by It? stochastic differential equation, the observation object equation has compound Poisson component, which allows simulating impulse noises and perturbations. Statistical algorithms are based on the reducing the filtration problem to the analysis of stochastic systems with terminating and branching paths by the interpretation of the term in Duncan–Mortensen– Zakai equation as an absorption and recovery function of sample paths for auxiliary random process. The solution of analysis problem can be found approximately by using numerical methods for solving stochastic differential equations and methods for modeling nonhomogeneous Poisson flows. The modeling algorithm for observation system and optimal estimation of its state based on the maximal section method is given in the paper. The main advantages of this algorithm are easy implementation and universality, namely the possibility of solving the optimal filtering problem for linear and non-linear models of the observation system, for one-dimensional and multidimensional case.