scholarly journals Statistical-mechanical analysis of adaptive filter with clipping saturation-type nonlinearity

2021 ◽  
Author(s):  
Seiji Miyoshi
Keyword(s):  
Steady State ◽  
Differential Equations ◽  
Asymptotic Analysis ◽  
Adaptive Filter ◽  
Adaptive System ◽  
Adaptive Signal Processing ◽  
Mean Square ◽  
Mean Square Stability ◽  
Statistical Mechanical ◽  
The Mean

Adaptive signal processing is used in broad areas. In most practical adaptive systems, there exists substantial nonlinearity that cannot be neglected. In this paper, we analyze the behaviors of an adaptive system in which the output of the adaptive filter has the clipping saturation-type nonlinearity by a statistical-mechanical method. To represent the macroscopic state of the system, we introduce two macroscopic variables. By considering the limit in which the number of taps of the unknown system and adaptive filter is large, we derive the simultaneous differential equations that describe the system behaviors in the deterministic and closed form. Although the derived simultaneous differential equations cannot be analytically solved, we discuss the dynamical behaviors and steady state of the adaptive system by asymptotic analysis, steady-state analysis, and numerical calculation. As a result, it becomes clear that the saturation value S has the critical value SC at which the mean-square stability of the adaptive system is lost. That is, when S > SC, both the mean-square error (MSE) and mean-square deviation (MSD) converge, i.e., the adaptive system is mean-square stable. On the other hand, when S < SC, the MSD diverges although the MSE converges, i.e., the adaptive system is not mean-square stable. In the latter case, the converged value of the MSE is a quadratic function of S and does not depend on the step size. Finally, SC is exactly derived by asymptotic analysis.<br>

scholarly journals Statistical-mechanical analysis of adaptive filter with clipping saturation-type nonlinearity

2021 ◽  
Author(s):  
Seiji Miyoshi
Keyword(s):  
Steady State ◽  
Differential Equations ◽  
Asymptotic Analysis ◽  
Adaptive Filter ◽  
Adaptive System ◽  
Adaptive Signal Processing ◽  
Mean Square ◽  
Mean Square Stability ◽  
Statistical Mechanical ◽  
The Mean

Adaptive signal processing is used in broad areas. In most practical adaptive systems, there exists substantial nonlinearity that cannot be neglected. In this paper, we analyze the behaviors of an adaptive system in which the output of the adaptive filter has the clipping saturation-type nonlinearity by a statistical-mechanical method. To represent the macroscopic state of the system, we introduce two macroscopic variables. By considering the limit in which the number of taps of the unknown system and adaptive filter is large, we derive the simultaneous differential equations that describe the system behaviors in the deterministic and closed form. Although the derived simultaneous differential equations cannot be analytically solved, we discuss the dynamical behaviors and steady state of the adaptive system by asymptotic analysis, steady-state analysis, and numerical calculation. As a result, it becomes clear that the saturation value S has the critical value SC at which the mean-square stability of the adaptive system is lost. That is, when S > SC, both the mean-square error (MSE) and mean-square deviation (MSD) converge, i.e., the adaptive system is mean-square stable. On the other hand, when S < SC, the MSD diverges although the MSE converges, i.e., the adaptive system is not mean-square stable. In the latter case, the converged value of the MSE is a quadratic function of S and does not depend on the step size. Finally, SC is exactly derived by asymptotic analysis.<br>

scholarly journals A Compensated Numerical Method for Solving Stochastic Differential Equations with Variable Delays and Random Jump Magnitudes

2015 ◽  
Vol 2015 ◽  
pp. 1-11
Author(s):  
Ying Du ◽  
Changlin Mei
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Numerical Method ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Mean Square ◽  
Variable Delays ◽  
Mean Square Stability ◽  
The Mean ◽  
Random Jump

Stochastic differential equations with jumps are of a wide application area especially in mathematical finance. In general, it is hard to obtain their analytical solutions and the construction of some numerical solutions with good performance is therefore an important task in practice. In this study, a compensated split-stepθmethod is proposed to numerically solve the stochastic differential equations with variable delays and random jump magnitudes. It is proved that the numerical solutions converge to the analytical solutions in mean-square with the approximate rate of 1/2. Furthermore, the mean-square stability of the exact solutions and the numerical solutions are investigated via a linear test equation and the results show that the proposed numerical method shares both the mean-square stability and the so-called A-stability.

scholarly journals On Mean Square Stability and Dissipativity of Split-Step Theta Method for Nonlinear Neutral Stochastic Delay Differential Equations

2016 ◽  
Vol 2016 ◽  
pp. 1-8
Author(s):  
Haiyan Yuan ◽  
Jihong Shen ◽  
Cheng Song
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Mean Square ◽  
Stochastic Delay Differential Equations ◽  
Stochastic Delay ◽  
Mean Square Stability ◽  
Linear Growth Condition ◽  
The Mean ◽  
Delay Differential ◽  
Theta Method

A split-step theta (SST) method is introduced and used to solve the nonlinear neutral stochastic delay differential equations (NSDDEs). The mean square asymptotic stability of the split-step theta (SST) method for nonlinear neutral stochastic delay differential equations is studied. It is proved that under the one-sided Lipschitz condition and the linear growth condition, the split-step theta method withθ∈(1/2,1]is asymptotically mean square stable for all positive step sizes, and the split-step theta method withθ∈[0,1/2]is asymptotically mean square stable for some step sizes. It is also proved in this paper that the split-step theta (SST) method possesses a bounded absorbing set which is independent of initial data, and the mean square dissipativity of this method is also proved.

scholarly journals Numerical Spline Method for Simulation of Stochastic Differential Equations systems: طريقة شرائحية عددية لمحاكاة حل نظم من المعادلات التفاضلية العشوائية

2021 ◽  
Vol 5 (4) ◽  
pp. 130-111
Author(s):  
Suliman M. Mahmoud, Ahmad Al-Wassouf, Ali S. Ehsaan Suliman M. Mahmoud, Ahmad Al-Wassouf, Ali S. Ehsaan
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Continuous Process ◽  
Mean Square ◽  
Spline Method ◽  
Third Order ◽  
Mean Square Stability ◽  
The Mean ◽  
Linear Problems ◽  
Two Parameters

In this paper, numerical spline method is presented with collocation two parameters for solving systems of multi-dimensional stochastic differential equations (SDEs). Multi-Wiener's time-continuous process is simulated as a discrete process, and then the mean-square stability of proposed method when applied to a system of two-dimensional linear SDEs is studied. The study shows that the method is mean-square stability and third-order convergent when applied to a system of linear and nonlinear SDEs. Moreover, the effectiveness of our method was tested by solving two test linear and non-linear problems. The numerical results show that the accuracy and applicability of the proposed method are worthy of attention.

scholarly journals Mean-Square Stability of Split-Step Theta Milstein Methods for Stochastic Differential Equations

2018 ◽  
Vol 2018 ◽  
pp. 1-13 ◽  
Author(s):  
Mahmoud A. Eissa ◽  
Haiying Zhang ◽  
Yu Xiao
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Mean Square ◽  
Step Size ◽  
Mean Square Stability ◽  
Test Equation ◽  
The Mean ◽  
The Stability ◽  
And Diffusion

The fundamental analysis of numerical methods for stochastic differential equations (SDEs) has been improved by constructing new split-step numerical methods. In this paper, we are interested in studying the mean-square (MS) stability of the new general drifting split-step theta Milstein (DSSθM) methods for SDEs. First, we consider scalar linear SDEs. The stability function of the DSSθM methods is investigated. Furthermore, the stability regions of the DSSθM methods are compared with those of test equation, and it is proved that the methods with θ≥3/2 are stochastically A-stable. Second, the nonlinear stability of DSSθM methods is studied. Under a coupled condition on the drifting and diffusion coefficients, it is proved that the methods with θ>1/2 can preserve the MS stability of the SDEs with no restriction on the step-size. Finally, numerical examples are given to examine the accuracy of the proposed methods under the stability conditions in approximation of SDEs.

scholarly journals Stability of a Class of Nonlinear Neutral Stochastic Differential Equations with Variable Time Delays

2012 ◽  
Vol 20 (1) ◽  
pp. 467-488 ◽  
Author(s):  
Meng Wu ◽  
Nanjing Huang ◽  
Changwen Zhao
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Time Delays ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Mean Square ◽  
Variable Time ◽  
Mean Square Stability ◽  
The Mean ◽  
Necessary And Sufficient

Abstract In this paper, we study the mean square asymptotic stability of a class of generalized nonlinear neutral stochastic differential equations with variable time delays by using fixed point theory. An asymptotic mean square stability theorem with a necessary and sufficient condition is proved which improves and generalizes some well-known results. Finally, two examples are given to illustrate our results.

scholarly journals Exponential Mean-Square Stability of Numerical Solutions to Stochastic Differential Equations

2003 ◽  
Vol 6 ◽  
pp. 297-313 ◽  
Author(s):  
Desmond J. Higham ◽  
Xuerong Mao ◽  
Andrew M. Stuart
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Numerical Method ◽  
Finite Time ◽  
Mean Square ◽  
Convergence Conditions ◽  
Finite Time Convergence ◽  
Mean Square Stability ◽  
Exponential Mean Square Stability ◽  
Positive Results

AbstractPositive results are proved here about the ability of numerical simulations to reproduce the exponential mean-square stability of stochastic differential equations (SDEs). The first set of results applies under finite-time convergence conditions on the numerical method. Under these conditions, the exponential mean-square stability of the SDE and that of the method (for sufficiently small step sizes) are shown to be equivalent, and the corresponding second-moment Lyapunov exponent bounds can be taken to be arbitrarily close. The required finite-time convergence conditions hold for the class of stochastic theta methods on globally Lipschitz problems. It is then shown that exponential mean-square stability for non-globally Lipschitz SDEs is not inherited, in general, by numerical methods. However, for a class of SDEs that satisfy a one-sided Lipschitz condition, positive results are obtained for two implicit methods. These results highlight the fact that for long-time simulation on nonlinear SDEs, the choice of numerical method can be crucial.

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