Filtering theory for a weakly coloured noise process

TL;DR

This paper develops a filtering theory for stochastic systems influenced by weakly coloured noise, extending classical methods to include processes like the Ornstein-Uhlenbeck process, and demonstrates its effectiveness through numerical simulations.

Contribution

It introduces a novel filtering approach for systems with weakly coloured noise, based on Stratonovich's perturbation theory and filtering density evolution.

Findings

Filtering method effectively estimates states in weakly coloured noise systems.

Numerical simulations confirm the accuracy of the proposed filtering algorithm.

The approach extends classical filtering to more realistic noise models.

Abstract

The problem of analyzing the Ito stochastic differential system and its filtering has received attention. The classical approach to accomplish filtering for the Ito SDE is the Kushner equation. In contrast to the classical filtering approach, this paper presents filtering for the stochastic differential system affected by weakly coloured noise. As a special case, the process can be regarded as the Ornstein-Uhlenbeck (OU) process. The theory of this paper is based on a pioneering contribution of Stratonovich involving the perturbation-theoretic approach to noisy dynamical systems in combination with the notion of the filtering density evolution. Making the use of the filtering density evolution equation, the stochastic evolution of condition moment is derived. A scalar Duffing system driven by the OU process is employed to test the effectiveness of the filtering theory of the paper. Numerical simulations involving four different sets of initial conditions and system parameters are utilized to examine the efficacy of the filtering algorithm of this paper.