On Asymptotic Stability of Stochastic Differential Equations with Delay in Infinite Dimensional Spaces

In most stochastic dynamical systems which describe process in engineering, physics and economics, stochastic components and random noise are often involved. Stochastic effects of these models are often used to capture the uncertainty about the operating systems. Motivated by the development of analysis and theory of stochastic processes, as well as the studies of natural sciences, the theory of stochastic differential equations in infinite dimensional spaces evolves gradually into a branch of modern analysis. In the analysis of such systems, we want to investigate their stabilities. This thesis is mainly concerned about the studies of the stability property of stochastic differential equations in infinite dimensional spaces, mainly in Hilbert spaces. Chapter 1 is an overview of the studies. In Chapter 2, we recall basic notations, definitions and preliminaries, especially those on stochastic integration and stochastic differential equations in infinite dimensional spaces. In this way, such notions as Q-Wiener processes, stochastic integrals, mild solutions will be reviewed. We also introduce the concepts of several types of stability. In Chapter 3, we are mainly concerned about the moment exponential stability of neutral impulsive stochastic delay partial differential equations with Poisson jumps. By employing the fixed point theorem, the p-th moment exponential stability of mild solutions to system is obtained. In Chapter 4, we firstly attempt to recall an impulsive-integral inequality by considering impulsive effects in stochastic systems. Then we define an attracting set and study the exponential stability of mild solutions to impulsive neutral stochastic delay partial differential equations with Poisson jumps by employing impulsive-integral inequality. Chapter 5 investigates p-th moment exponential stability and almost sure asymptotic stability of mild solutions to stochastic delay integro-differential equations. Finally in Chapter 6, we study the exponential stability of neutral impulsive stochastic delay partial differential equations driven by a fractional Brownian motion

Asymptotic stability of stochastic differential equations driven by Lévy noise

Using key tools such as Ito's formula for general semimartingales, Kunita's moment estimates for Levy-type stochastic integrals, and the exponential martingale inequality, we find conditions under which the solutions to the stochastic differential equations (SDEs) driven by Levy noise are stable in probability, almost surely and moment exponentially stable

Convergence rate of numerical solutions to SFDEs with jumps

In this paper, we are interested in numerical solutions of stochastic functional differential equations with jumps. Under a global Lipschitz condition, we show that the pth-moment convergence of Euler–Maruyama numerical solutions to stochastic functional differential equations with jumps has order 1/p for any p ≥ 2. This is significantly different from the case of stochastic functional differential equations without jumps, where the order is 1/2 for any p ≥ 2. It is therefore best to use the mean-square convergence for stochastic functional differential equations with jumps. Moreover, under a local Lipschitz condition, we reveal that the order of mean-square convergence is close to 1/2, provided that local Lipschitz constants, valid on balls of radius j, do not grow faster than log j

Asymptotic stability of stochastic differential equations driven by Lévy noise

Using key tools such as Ito's formula for general semimartingales, Kunita's moment estimates for Levy-type stochastic integrals, and the exponential martingale inequality, we find conditions under which the solutions to the stochastic differential equations (SDEs) driven by Levy noise are stable in probability, almost surely and moment exponentially stable

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)

Competitive Lotka-Volterra Population Dynamics with Jumps

This paper considers competitive Lotka-Volterra population dynamics with jumps. The contributions of this paper are as follows. (a) We show stochastic differential equation (SDE) with jumps associated with the model has a unique global positive solution; (b) We discuss the uniform boundedness of $p$th moment with $p>0$ and reveal the sample Lyapunov exponents; (c) Using a variation-of-constants formula for a class of SDEs with jumps, we provide explicit solution for 1-dimensional competitive Lotka-Volterra population dynamics with jumps, and investigate the sample Lyapunov exponent for each component and the extinction of our $n$-dimensional model.Comment: 25 page

On almost sure stability of hybrid stochastic systems with mode-dependent interval delays

This note develops a criterion for almost sure stability of hybrid stochastic systems with mode-dependent interval time delays, which improves an existing result by exploiting the relation between the bounds of the time delays and the generator of the continuous-time Markov chain. The improved result shows that the presence of Markovian switching is quite involved in the stability analysis of delay systems. Numerical examples are given to verify the effectiveness