Asymptotic mean-square stability of two-step methods for stochastic ordinary differential equations

We deal with linear multi-step methods for SDEs and study when the numerical appro\-xi\-mation shares asymptotic properties in the mean-square sense of the exact solution. As in deterministic numerical analysis we use a linear time-invariant test equation and perform a linear stability analysis. Standard approaches used either to analyse deterministic multi-step methods or stochastic one-step methods do not carry over to stochastic multi-step schemes. In order to obtain sufficient conditions for asymptotic mean-square stability of stochastic linear two-step-Maruyama methods we construct and apply Lyapunov-type functionals. In particular we study the asymptotic mean-square stability of stochastic counterparts of two-step Adams-Bashforth- and Adams-Moulton-methods, the Milne-Simpson method and the BDF method

Asymptotic mean square stability analysis for a stochastic delay differential equation

For a stochastic delay differential equation, the effects of noise and time delay are discussed in the sense of mean square stability. Neither time delay nor noise play bad roles for the differential equations and both of them are ubiquitous in nature. The so-called domain subdivision approach is taken to study the stability regions in terms of the parameters of a given equation and the Ito formula is employed to deal with the fluctuation noise. An interesting result demonstrated in this paper shows that noise with appropriate power could reduce the influence of time delay. © 2011 SICE

Устойчивость в среднем квадратичном решений автономных диффузионных динамических систем с конечным последействием с учетом случайных факторов

Отримано необхідні та достатні умови асимптотичної стійкості у середньому квадратичному сильного розв’язку автономного дифузійного стохастичного диференціально-функціонального рівняння з кінцевою післядією з врахуванням випадкових факторів (дія на систему ззовні випадкових функцій з різними законами розподілу).The necessary and sufficient conditions are obtained for the asymptotic mean square stability of strong solutions of autonomous diffusion stochastic functional-differential equations with finite after-effect and random factors (random functions with different distribution) taken into account

Unifying Dynamical and Structural Stability of Equilibriums

We exhibit a fundamental relationship between measures of dynamical and structural stability of equilibriums, arising from real dynamical systems. We show that dynamical stability, quantified via systems local response to external perturbations, coincides with the minimal internal perturbation able to destabilize the equilibrium. First, by reformulating a result of control theory, we explain that harmonic external perturbations reflect the spectral sensitivity of the Jacobian matrix at the equilibrium, with respect to constant changes of its coefficients. However, for this equivalence to hold, imaginary changes of the Jacobian's coefficients have to be allowed. The connection with dynamical stability is thus lost for real dynamical systems. We show that this issue can be avoided, thus recovering the fundamental link between dynamical and structural stability, by considering stochastic noise as external and internal perturbations. More precisely, we demonstrate that a system's local response to white-noise perturbations directly reflects the intensity of internal white noise that it can accommodate before asymptotic mean-square stability of the equilibrium is lost.Comment: 13 pages, 2 figure

Mean-square stability analysis of approximations of stochastic differential equations in infinite dimensions

The (asymptotic) behaviour of the second moment of solutions to stochastic differential equations is treated in mean-square stability analysis. This property is discussed for approximations of infinite-dimensional stochastic differential equations and necessary and sufficient conditions ensuring mean-square stability are given. They are applied to typical discretization schemes such as combinations of spectral Galerkin, finite element, Euler-Maruyama, Milstein, Crank-Nicolson, and forward and backward Euler methods. Furthermore, results on the relation to stability properties of corresponding analytical solutions are provided. Simulations of the stochastic heat equation illustrate the theory.Comment: 22 pages, 4 figures; deleted a section; shortened the presentation of results; corrected typo

Spectral Perspective on the Stability of Discrete-Time Markov Jump Systems with Multiplicative Noise

We apply the spectrum analysis approach to address the stability of discrete-time Markov jump systems with state-multiplicative noise. In terms of the spectral distribution of a generalized Lyapunov operator, spectral criteria are presented to testify three different kinds of stochastic stabilities: asymptotic mean square stability, critical stability, and essential instability

A comparative linear mean-square stability analysis of Maruyama- and Milstein-type methods

In this article we compare the mean-square stability properties of the Theta-Maruyama and Theta-Milstein method that are used to solve stochastic differential equations. For the linear stability analysis, we propose an extension of the standard geometric Brownian motion as a test equation and consider a scalar linear test equation with several multiplicative noise terms. This test equation allows to begin investigating the influence of multi-dimensional noise on the stability behaviour of the methods while the analysis is still tractable. Our findings include: (i) the stability condition for the Theta-Milstein method and thus, for some choices of Theta, the conditions on the step-size, are much more restrictive than those for the Theta-Maruyama method; (ii) the precise stability region of the Theta-Milstein method explicitly depends on the noise terms. Further, we investigate the effect of introducing partially implicitness in the diffusion approximation terms of Milstein-type methods, thus obtaining the possibility to control the stability properties of these methods with a further method parameter Sigma. Numerical examples illustrate the results and provide a comparison of the stability behaviour of the different methods.Comment: 19 pages, 10 figure

Robust H

The problem of robust H∞ filter design is investigated for stochastic pantograph systems governed by linear Itô differential equation. First, a sufficient condition for asymptotic mean-square stability of stochastic pantograph systems is presented by means of Lyapunov approach. Then, based on matrix inequalities, the H∞ filtering problem for this kind of systems is studied and a sufficient condition for the existence of the H∞ filter is derived. Furthermore, the explicit expression of the desired filter parameters is characterized. Finally, an example is given to illustrate the results

Asymptotic mean-square stability analysis and simulations of a stochastic model for the human immune response with memory

In this thesis we extended a deterministic model for the Human Immune response, consisting of a non-linear system of differential equations with distributed time delay, which was introduced by Beretta, Kirshner and Marino in 2007, by incorporating stochastic perturbations with multiplicative noise around equilibria of the deterministic model. Our aim is to study the robustness of the equilibria of the deterministic model for the human immune response system with respect to fluctuations due to considering the human body as a noisy enviroment. We do this by analysing the asymptotic mean square stability of the equilibria of our stochastic model. Our work could be divided roughly into two parts. In the first part we analyse the stability of a general non linear system of stochastic differential equations with distributed memory terms by studying the stability properties of the linearisation in the first approximation. First of all we state, using Halanay's inequalities, comparison results useful in the investigation of exponential mean square stability of linear stochastic delay differential systems with distributed memory terms. Then we provide conditions under which asymptotic mean square stability of a nonlinear system of stochastic delay differential equations is implied by the exponential mean square stability of linearised stochastic delay system in the first approximation. In the second part we apply the theoretical results obtained in the first part to investigate the stochastic stability properties of the equilibria of our stochastic model of human immune response. The theoretical results are illustrated by numerical simulations and an uncertainty and sensivity analysis of our stochastic model, suggesting that the deterministic model is robust with respect to the stochastic perturbations