Exponential stability of Euler-Maruyama solutions for impulsive stochastic differential equations with delay

This paper establishes a method to study the exponential stability of Euler-Maruyama (EM) method for impulsive stochastic differential equations with delay. By using the properties of M-matrix and stochastic analysis technique, some conditions under which the EM solution is exponentially mean-square stable are obtained. Some examples are provided to illustrate the results.     

Asymptotic properties of neutral stochastic differential delay equations

This paper discusses asymptotic properties, especially asymptotic stability of neutral stochastic differential delay equations. New techniques are developed to cope with the neutral delay case, and the results of this paper are more general than the author's earlier work within the delay equations     

矩阵微分方程的等度有界性

用矩阵李雅普诺夫函数研究了矩阵微分方程的等度有界性,给出了非自治矩阵微分方程等度有界性的几个判定定理,实例说明了主要定理的实用性     

Almost sure exponential stability of backward Euler-Maruyama discretizations for hybrid stochastic differential equations

This is a continuation of the first author’s earlier paper [1] jointly with Pang and Deng, in which the authors established some sufficient conditions under which the Euler-Maruyama (EM) method can reproduce the almost sure exponential stability of the test hybrid SDEs. The key condition imposed in [1] is the global Lipschitz condition. However, we will show in this paper that without this global Lipschitz condition the EM method may not preserve the almost sure exponential stability. We will then show that the backward EM method can capture the almost sure exponential stability for a certain class of highly nonlinear hybrid SDEs.     

多维马尔科夫转制随机微分方程的数值解

由于多维马尔科夫转制随机微分方程不存在解析解,利用Euler—Maruyama方法给出多维马尔科夫转制随机微分方程的渐进数值解,并证明了此数值解收敛到方程的解析解．将单一马尔科夫转制随机微分方程的数值解问题延伸到多维马尔科夫转制情形,增强了马尔科夫转制随机微分方程的适用性．     

Mean-square exponential dichotomy of numerical solutions to stochastic differential equations

We present the ability of numerical simulations to reproduce the mean-square exponential dichotomy of stochastic differential equations. Under some conditions, we show that the mean-square exponential dichotomy of stochastic differential equations is equivalent to that of the numerical method for sufficient small step sizes     

STABILITY OF FUNCTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAYS

In this paper,the stability of a class of impulsive functional differential equations with infinite delays is investigated. A uniform stability theorem and a uniform asymptotic stability theorem are established.     

On the practical global uniform asymptotic stability of stochastic differential equations

The method of Lyapunov functions is one of the most effective ones for the investigation of stability of dynamical systems, in particular, of stochastic differential systems. The main purpose of the paper is the analysis of the stability of stochastic differential equations (SDEs) by using Lyapunov functions when the origin is not necessarily an equilibrium point. The global uniform boundedness and the global practical uniform exponential stability of solutions of SDEs based on Lyapunov techniques are investigated. Furthermore, an example is given to illustrate the applicability of the main result.     

The p-th moment stability of solutions to impulsive stochastic differential equations driven by G-Brownian motion

In this paper, we consider a class of impulsive stochastic differential equations driven by G-Brownian motion (IGSDEs in short). By means of the G-Lyapunov function method, some criteria on p-th moment stability and p-th moment asymptotical stability for the trivial solutions of IGSDEs are established. An example is presented to illustrate the efficiency of the obtained results.     

Stationarity and stochastic differential equations

A linear differential equation of order N with stochastic process coefficients and excitation are studied. The objective of this paper is to demonstrate that, by using an expansion method, when the coefficients and excitation are strict sense stationary processes, the response is also a strict sense stationary process. Such problems occur frequently in the engineering sciences and are very important. Applications include parametric random vibrations, turbulent environment rotorcraft dynamics, and dynamics of axially loaded structural members, among others. An example application is provided.     

Numerical methods for nonlinear stochastic differential equations with jumps

We present and analyse two implicit methods for Ito stochastic differential equations (SDEs) with Poisson-driven jumps. The first method, SSBE, is a split-step extension of the backward Euler method. The second method, CSSBE, arises from the introduction of a compensated, martingale, form of the Poisson process. We show that both methods are amenable to rigorous analysis when a one-sided Lipschitz condition, rather than a more restrictive global Lipschitz condition, holds for the drift. Our analysis covers strong convergence and nonlinear stability. We prove that both methods give strong convergence when the drift coefficient is one-sided Lipschitz and the diffusion and jump coefficients are globally Lipschitz. On the way to proving these results, we show that a compensated form of the Euler–Maruyama method converges strongly when the SDE coefficients satisfy a local Lipschitz condition and the pth moment of the exact and numerical solution are bounded for some p>2. Under our assumptions, both SSBE and CSSBE give well-defined, unique solutions for sufficiently small stepsizes, and SSBE has the advantage that the restriction is independent of the jump intensity. We also study the ability of the methods to reproduce exponential mean-square stability in the case where the drift has a negative one-sided Lipschitz constant. This work extends the deterministic nonlinear stability theory in numerical analysis. We find that SSBE preserves stability under a stepsize constraint that is independent of the initial data. CSSBE satisfies an even stronger condition, and gives a generalization of B-stability. Finally, we specialize to a linear test problem and show that CSSBE has a natural extension of deterministic A-stability. The difference in stability properties of the SSBE and CSSBE methods emphasizes that the addition of a jump term has a significant effect that cannot be deduced directly from the non-jump literature.This work was supported by Engineering and Physical Sciences Research Council grant GR/T19100 and by a Research Fellowship from The Royal Society of Edinburgh/Scottish Executive Education and Lifelong Learning Department.     

Preserving positivity in solutions of discretised stochastic differential equations

We consider the Euler discretisation of a scalar linear test equation with positive solutions and show for both strong and weak approximations that the probability of positivity over any finite interval of simulation tends to unity as the step size approaches zero. Although a.s. positivity in an approximation is impossible to achieve, we develop for the strong (Maruyama) approximation an asymptotic estimate of the number of mesh points required for positivity as our tolerance of non-positive trajectories tends to zero, and examine the effectiveness of this estimate in the context of practical numerical simulation. We show how this analysis generalises to equations with a drift coefficient that may display a high level of nonlinearity, but which must be linearly bounded from below (i.e. when acting towards zero), and a linearly bounded diffusion coefficient. Finally, in the linear case we develop a refined asymptotic estimate that is more useful as an a priori guide to the number of mesh points required to produce positive approximations with a given probability.     

随机泛函微分方程正整体解的存在唯一性及矩有界性

构造了Lyapunov函数V (x),然后给出一个一般条件,应用Khasminskii?Mao定理,得到非线性随机泛函微分方程(SFDEs)存在正整体解,且这个解p阶矩有界.     

Strict stability of impulsive functional differential equations

Strict stability is the kind of stability that can give us some information about the rate of decay of the solutions. There are some results about strict stability of differential equations. In the present paper, we shall extend the strict stability to impulsive functional differential equations. By using Lyapunov functions and Razumikhin technique, we shall get some criteria for the strict stability of impulsive functional differential equations, and we can see that impulses do contribute to the system's strict stability behavior.     

Stability analysis of highly nonlinear hybrid multiple-delay stochastic differential equations

Stability criteria for stochastic differential delay equations (SDDEs) have been studied intensively for the past few decades. However, most of these criteria can only be applied to delay equations where their coefficients are either linear or nonlinear but bounded by linear functions. Recently, the stability of highly nonlinear hybrid stochastic differential equations with a single delay is investigated in [Fei, Hu, Mao and Shen, Automatica, 2017], whose work, in this paper, is extended to highly nonlinear hybrid stochastic differential equations with variable multiple delays. In other words, this paper establishes the stability criteria of highly nonlinear hybrid variable multiple-delay stochastic differential equations. We also discuss an example to illustrate our results.     

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??A class of backward doubly stochastic differential equations driven by white noises and Poisson random measures are studied in this paper. The definitions of solutions and Yamada-Watanabe type theorem to this equation are established.