Complete Real Time Solution of the General Nonlinear Filtering Problem without Memory

TL;DR

This paper presents a real-time, memoryless algorithm for solving the general nonlinear filtering problem by extending previous methods, with proven convergence, error estimates, and numerical validation.

Contribution

It extends existing algorithms to handle the most general nonlinear filtering cases with explicit time dependence and matrix-valued noise variances.

Findings

Algorithm converges to the true solution in L^1 sense.

Error estimates for the approximation are provided.

Numerical simulations demonstrate the algorithm's feasibility and efficiency.

Abstract

It is well known that the nonlinear filtering problem has important applications in both military and civil industries. The central problem of nonlinear filtering is to solve the Duncan-Mortensen-Zakai (DMZ) equation in real time and in a memoryless manner. In this paper, we shall extend the algorithm developed previously by S.-T. Yau and the second author to the most general setting of nonlinear filterings, where the explicit time-dependence is in the drift term, observation term, and the variance of the noises could be a matrix of functions of both time and the states. To preserve the off-line virture of the algorithm, necessary modifications are illustrated clearly. Moreover, it is shown rigorously that the approximated solution obtained by the algorithm converges to the real solution in the $L^1$ sense. And the precise error has been estimated. Finally, the numerical simulation support the feasibility and efficiency of our algorithm.