To construction of an optimal filter for random processes

Abstract

We consider the generalized integral equation of optimal filter of Wiener- Kolmogorov for nonstationary random processes. Solvability and construction of the general solution of a generalized integral equation remains an unsolved problem. In this paper we propose a method for solving an integral equation in the weight matrix. A necessary and sufficient condition for the existence of a solution of the integral equation is obtained. General solution of the integral equation is found. The case where the desired random process which is a solution of the stochastic differential equation, and the equation of the optimal filter is linear equations with unknown matrices. The parameters of the optimal linear filter are defined. A new method of constructing an optimal filter for diffusion processes is supposed. Arose from the need to practice optimal filters Kalman - Bucy are one of the best results in the theory of optimal filtering. However, the solution of the matrix Riccati equation is difficult, and it can be solved only by approximate methods. Therefore it is interested to develop a new method for the optimal filtering of diffusion processes.