Filtering Theory for a Weakly Coloured Noise Process

Abstract

The problem of analyzing the Itô stochastic differential system and its filtering has received attention. The classical approach to accomplish filtering for the Itô 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, i.e., $\dot{x}_{t} = f\left( {x_{t} } \right) + g\left( {x_{t} } \right)\xi_{t} ,$ where the input process $\xi_{t}$ is a weakly coloured noise process. As a special case, the process $\xi_{t}$ can be regarded as the Ornstein–Uhlenbeck (OU) process, i.e., $d\xi_{t} = - \alpha \xi_{t} dt + \beta dB_{t} ,$ where $\alpha > 0.$ More precisely, the filtering model of this paper can be cast as

$\dot{x}_{t} = f\left( {x_{t} } \right) + g\left( {x_{t} } \right)\xi_{t} ,$

$z_{t} = \int\limits_{{t_{0} }}^{t} {h(x_{\tau } )d\tau + \eta_{t} ,}$

where $h(x_{t} )$ is the measurement non-linearity and $\eta = \{ \eta_{t} ,t_{0} \le t < \infty \}$ is the Brownian motion process. The former expression describes the structure of a noisy dynamical system, and the latter is the observation equation. The novelties of this paper are two (1) the extension of the filtering theory for the Itô stochastic differential system to the filtering theory for the ‘weakly coloured noise-driven’ stochastic differential system (2) the theory of this paper is based on a pioneering contribution of Ruslan Stratonovich involving the perturbation-theoretic approach to noisy dynamical systems in combination with the notion of the ‘filtering density’ evolution. The stochastic evolution of condition moment is derived by utilizing the filtering density evolution equation. 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.

Appendix A: The Fokker–Planck Equation

The Fokker–Planck equation is the evolution of conditional probability density for given initial states for a Markov process. The evolution of the conditional probability density of the non-Markov process is known as the stochastic equation. The stochastic equation for a non-Markov process, which satisfies $\dot{x}_{t} = f(x_{t} ) + g(x_{t} )\xi_{t} ,$ where the input process $\xi_{t}$ has relatively smaller correlation time, reduces to the Fokker–Planck equation. Consider the input process $\xi_{t}$ is OU process with smaller correlation time, the Fokker–Planck equation can be derived using the functional calculus [8]. Here, we state an alternative and a brief approach to arrive at the Fokker–Planck equation for the OU process-driven stochastic differential system. The standard structure of the Fokker–Planck equation is given in Stratonovich ([22], p. 62), i.e.,and consider

$$\dot{p}\left( x \right) = - \frac{\partial }{\partial x}k_{1} \left( x \right)p\left( x \right) + \frac{1}{2}\frac{{\partial^{2} k_{2} \left( x \right)p\left( x \right)}}{{\partial x^{2} }},$$

$$k_{1} \left( x \right) = M\left( x \right) + \frac{1}{4}\frac{dk\left( x \right)}{{dx}},$$

(29)

$$k_{2} \left( x \right) = k\left( x \right),$$

(30)

a simple calculation shows that the Fokker–Planck equation can be recast as

$$\dot{p}\left( x \right) = - \frac{\partial }{\partial x}M\left( x \right)p\left( x \right) + \frac{1}{4}\frac{\partial }{\partial x}\left( {k\left( x \right)\frac{\partial p}{{\partial x}} + \frac{\partial k\left( x \right)p\left( x \right)}{{\partial x}}} \right).$$

(31)

The terms $k_{1} (x)$ and $k_{2} (x)$ for the OU process-driven stochastic differential system are

$$k_{1} (x) = f + Dgg^{\prime} + D\tau_{cor} g^{2} g^{\prime}\left( \frac{f}{g} \right)^{\prime },$$

(32)

$$k_{2} (x) = 2Dg^{2} + 2D\tau_{cor} g^{3} \left( \frac{f}{g} \right)^{\prime }.$$

(33)

Equations (4, 5) in combination with Eq. (10) result eq. (32, 33). Equations (32, 33) in conjunction with eq. (29, 30) lead to

$$\begin{aligned} k(x) & = 2Dg^{2} + 2D\tau_{cor} g^{3} \left( \frac{f}{g} \right)^{\prime }, \\ \frac{1}{4}\frac{\partial k(x)}{{\partial x}} = Dgg^{\prime} + \frac{3}{2}D\tau_{cor} g^{2} g^{\prime}\left( \frac{f}{g} \right)^{\prime } + \frac{1}{2}D\tau_{cor} g^{3} \left( \frac{f}{g} \right)^{\prime \prime }, \\ M(x) = f - \frac{1}{2}D\tau_{cor} g^{2} g^{\prime}\left( \frac{f}{g} \right)^{\prime } - \frac{1}{2}D\tau_{cor} g^{3} \left( \frac{f}{g} \right)^{\prime\prime }. \\ \end{aligned}$$

(34)

Equation (31) in combination with Eq. (34) leads to

$$\begin{aligned} \dot{p}(x) & = - \frac{\partial }{\partial x}\left( {f - \frac{{D\tau_{cor} }}{2}g^{2} g^{\prime}\left( \frac{f}{g} \right)^{\prime } - \frac{1}{2}D\tau_{cor} g^{3} \left( \frac{f}{g} \right)^{\prime\prime }} \right)p + \frac{1}{4}\frac{\partial }{\partial x}\left( {\left( {2Dg^{2} + 2D\tau_{cor} g^{3} \left( \frac{f}{g} \right)^{\prime }} \right)\frac{\partial p}{{\partial x}}} \right. \\ \left. { + \frac{\partial }{\partial x}\left( {2Dg^{2} + 2D\tau_{cor} g^{3} \left( \frac{f}{g} \right)^{\prime }} \right)p} \right). \\ \end{aligned}$$

Further simplification leads to

$$\begin{aligned} \dot{p}(x) & = - \frac{\partial }{\partial x}fp + \frac{{D\tau_{cor} }}{2}\frac{\partial }{\partial x}g^{2} g^{\prime}\left( \frac{f}{g} \right)^{\prime }p + \frac{1}{2}D\tau_{cor} \frac{\partial }{\partial x}g^{3} \left( \frac{f}{g} \right)^{\prime\prime }p \\ + \frac{1}{4}\frac{\partial }{\partial x}\left( {\left( {2Dg^{2} + 2D\tau_{cor} g^{3} \left( \frac{f}{g} \right)^{\prime }} \right)\frac{\partial p}{{\partial x}}} \right) + \frac{1}{4}\frac{\partial }{\partial x}\left( {\left( {2Dg^{2} + 2D\tau_{cor} g^{3} \left( \frac{f}{g} \right)^{\prime }} \right)\frac{\partial p}{{\partial x}}} \right) \\ + \frac{1}{4}\frac{\partial }{\partial x}\left( {4Dgg^{\prime} + 2D\tau_{cor} g^{3} \left( \frac{f}{g} \right)^{\prime\prime } + 6D\tau_{cor} g^{2} g^{\prime}\left( \frac{f}{g} \right)^{\prime }} \right)p. \\ \end{aligned}$$

A re-arrangement of the right-hand side terms of the above equation results

$$\dot{p}(x) = - \frac{\partial }{\partial x}fp + \frac{\partial }{\partial x}\left( {\left( {Dg^{2} + D\tau_{cor} g^{3} \left( \frac{f}{g} \right)^{\prime }} \right)\frac{\partial p}{{\partial x}}} \right) + \frac{\partial }{\partial x}\left( {Dgg^{\prime} + D\tau_{cor} g^{3} \left( \frac{f}{g} \right)^{\prime\prime } + 2D\tau_{cor} g^{2} g^{\prime}\left( \frac{f}{g} \right)^{\prime }} \right)p,$$

Furthermore,after combining the last two terms of the above equation, we have

$$\dot{p}(x) = - \frac{\partial }{\partial x}fp + D\frac{\partial }{\partial x}g\left( {\left( {g + \tau_{cor} g^{2} \left( \frac{f}{g} \right)^{\prime }} \right)\frac{\partial p}{{\partial x}}} \right) + D\frac{\partial }{\partial x}g\left( {g^{\prime} + \tau_{cor} g^{2} \left( \frac{f}{g} \right)^{\prime\prime } + 2\tau_{cor} gg^{\prime}\left( \frac{f}{g} \right)^{\prime }} \right)p,$$

$$\dot{p}(x) = - \frac{\partial }{\partial x}fp + D\frac{\partial }{\partial x}\left( {g\frac{\partial }{\partial x}\left( {g + \tau_{cor} g^{2} \left( \frac{f}{g} \right)^{\prime }} \right)p} \right) = - \frac{\partial }{\partial x}fp + D\frac{\partial }{\partial x}\left( {g\frac{\partial }{\partial x}g\left( {1 + \tau_{cor} g\left( \frac{f}{g} \right)^{\prime }} \right)p} \right),$$

(35)

Eq. (35) can be regarded as the Fokker–Planck equation for the OU process-driven stochastic differential system, where the OU process is a weakly coloured noise process.