Research On Exponential Stability Of Several Kinds Of Stochastic Differential Equations

Stochastic differential equations(SDEs)have been widely used in science and indus-try,such as finance,quantity economy,neural networks,biology,control etc.Moreover,the real systems are inevitably affected by stochastic perturbations resulting in instabili-ty,therefore,it is very necessary to study the stability of SDEs.In the study of stochastic stability,Lyapunov functions technique is one of the most classical and powerful tech-niques.However,in general,how to construct an appropriate Lyapunov function is a very difficult problem.In the absence of a suitable Lyapunov function,employing numerical methods to investigate the stability of SDEs is essential and it has become an important method to study the stochastic stability.In this dissertation,the relationship between the exponential stability of exact solutions and numerical solutions to three kinds of S-DEs(SDEs with Markov switching?SDEs with Markov switching and Poisson jumps?SDEs driven by G-Brown motion)is discussed.Under some conditions,this dissertation establishes the theory that if the SDE is exponentially stable,then the numerical method is exponentially stable,and if the numerical method is exponentially stable,then the SDE is exponentially stable.Therefore,the necessary and sufficient conditions of exponential stability between the SDE and the numerical method are obtained.And then,employing numerical methods to study the stability of SDEs instead of Lyapunonv functions tech-nique provides another way to study the stability of SDEs.The special research works are as follows.In Chapter 1,the research background of this dissertation,the development of ex-ponential stability in the numerical simulation of stochastic differential equations,basic knowledge of stochastic analysis and G-Brown motion are briefly introduced.In Chapter 2,when p ?(0,1),we mainly study that the SDE with Markov switching is p-th moment exponentially stable if and only if the split-step stochastic 0-method with sufficiently small step-size is p-th moment exponentially stable under the global Lipschitz condition.This chapter also shows that the p-th moment exponential stability of the SDE with Markov switching or the split-step stochastic 0-method implies the almost sure exponential stability of the SDE with Markov switching or the split-step stochastic 0-method.And then,under some conditions,whether such equation is p-th moment exponentially stable or the split-step stochastic 0-method,we can infer that the SDE with Markov switching is almost surely exponentially stable and the split-step stochastic 0-method.Finally,numerical examples are presented to support the correctness of the obtained theoretical results.In Chapter 3,when p ?(0,1),we consider that the SDE with Markov switching and Poisson jumps is p-th moment exponentially stable if and only if the split-step stochastic 0-method with sufficiently small step-size is p-th moment exponentially stable under the global Lipschitz condition.This chapter also illustrates that the p-th moment exponential stability of the SDE with Markov switching and Poisson jumps or the split-step stochastic 0-method implies the almost sure exponential stability of the SDE with Markov switch-ing and Poisson jumps or the split-step stochastic 0-method.And then,under certain conditions,whether such equation is p-th moment exponentially stable or the split-step stochastic 0-method,we can obtain that the SDE with Markov switching and Poisson jumps is almost surely exponentially stable and the split-step stochastic 0-method.Fi-nally,numerical simulations are given to show the efficiency of the obtained theoretical results.In Chapter 4,when p ?(0,1),we show that the SDE driven by G-Brown motion is p-th moment exponentially stable if and only if the split-step stochastic 0-method with sufficiently small step-size is p-th moment exponentially stable under the global Lipschitz condition.This chapter also proves that the p-th moment exponential stability of the SDE driven by G-Brown motion or the split-step stochastic 0-method implies the quasi sure exponential stability of the SDE driven by G-Brown motion or the split-step stochastic 0-method.And then,under some conditions,whether such equation is p-th moment exponentially stable or the split-step stochastic 0-method,we can infer that the SDE driven by G-Brown motion is quasi surely exponentially stable and the split-step stochastic 0-method.Finally,numerical experiments are provided to present the efficiency of the obtained theoretical results.