Stability of stochastic impulsive differential equations: integrating the cyber and the physical of stochastic systems

According to Newton's second law of motion, we humans describe a dynamical system with a differential equation, which is naturally discretized into a difference equation whenever a computer is used. The differential equation is the physical model in human brains and the difference equation the cyber model in computers for the dynamical system. The physical model refers to the dynamical system itself (particularly, a human-designed system) in the physical world and the cyber model symbolises it in the cyber counterpart. This paper formulates a hybrid model with impulsive differential equations for the dynamical system, which integrates its physical model in real world/human brains and its cyber counterpart in computers. The presented results establish a theoretic foundation for the scientific study of control and communication in the animal/human and the machine (Norbert Wiener) in the era of rise of the machines as well as a systems science for cyber-physical systems (CPS)

Mean-square convergence and stability of the backward Euler method for stochastic differential delay equations with highly nonlinear growing coefficients

Over the last few decades, the numerical methods for stochastic differential delay equations (SDDEs) have been investigated and developed by many scholars. Nevertheless, there is still little work to be completed. By virtue of the novel technique, this paper focuses on the mean-square convergence and stability of the backward Euler method (BEM) for SDDEs whose drift and diffusion coefficients can both grow polynomially. The upper mean-square error bounds of BEM are obtained. Then the convergence rate, which is one-half, is revealed without using the moment boundedness of numerical solutions. Furthermore, under fairly general conditions, the novel technique is applied to prove that the BEM can inherit the exponential mean-square stability with a simple proof. At last, two numerical experiments are implemented to illustrate the reliability of the theories

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

Positive results are derived concerning the long time dynamics of numerical simulations of stochastic differential equation systems with Markovian switching. Euler-Maruyama discretizations are shown to capture almost sure and momente xponential stability for all sufficiently small timesteps under appropriate conditions

The split-step backward Euler method for linear stochastic delay differential equations

AbstractIn this paper, the numerical approximation of solutions of linear stochastic delay differential equations (SDDEs) in the Itô sense is considered. We construct split-step backward Euler (SSBE) method for solving linear SDDEs and develop the fundamental numerical analysis concerning its strong convergence and mean-square stability. It is proved that the SSBE method is convergent with strong order γ=12 in the mean-square sense. The conditions under which the SSBE method is mean-square stable (MS-stable) and general mean-square stable (GMS-stable) are obtained. Some illustrative numerical examples are presented to demonstrate the order of strong convergence and the mean-square stability of the SSBE method

Convergence Rate of EM Scheme for SDDEs

In this paper we investigate the convergence rate of Euler-Maruyama scheme for a class of stochastic differential delay equations, where the corresponding coefficients may be highly nonlinear with respect to the delay variables. In particular, we reveal that the convergence rate of Euler-Maruyama scheme is 1/2$ for the Brownian motion case, while show that it is best to use the mean-square convergence for the pure jump case, and that the order of mean-square convergence is close to 1/2.Comment: Page 1