Research On The Stability Of Several Kinds Of Neutral Stochastic Delay Differential Equations With Markov Switching

The theory of stochastic differential equation is widely used in finance,system science,engineering science and other fields.For example,in the financial field,it can be used to solve the option pricing problem;in the biological field,it can be used to reveal the occurrence law of diseases;in the physical field,it can be used to study the escape and transition of Brown particles.The development and changes of the dynamical system in reality except for the influence of random factors,not only is related to the current state and the state of the past,but also is associated with the rate of change of the past state.Meanwhile,these dynamical systems may exchange between finite states because of the internal and external changes.Neutral stochastic delay differential equations with switching can well describe such hybrid systems,so more and more scholars pay attention to study the neutral stochastic delay differential equations with Markov switching.The stability theory plays an important role in stochastic dynamical systems,the pth moment exponential stability and almost surely exponential stability of the trivial solution are one of the hot topics.Although there have been many research results on the stability of the neutral stochastic delay differential equation model with Markov switching,there are still some unsolved problems.In this thesis,we study the following two aspects in the stability of neutral stochastic delay differential equations with Markov switching:On the one hand,the p(0<p?2)th moment exponential stability and the p(0<p?2)th moment boundedness of the neutral stochastic delay differential equations with Markov switching are studied.Based on the basic inequalities,integral inequalities and semi-martingale convergence theorems,we establish a new criterion for p(0<p?2)th moment exponential stability and the p(0<p?2)th bounded moment of the neutral stochastic delay differential equation with Markov switching when the delay term is continuous differentiable function.Our results are the supplement to the existing situation when p?2.And some examples are given to illustrate the obtained results.On the other hand,we obtain a new criterion for the existence uniqueness and almost surely exponential stability p(p?2)of global solutions of neutral stochastic delay differential equations with Markov switching and Poisson jump,and give an example is to illustrate the effectiveness of the obtained results.On the basis of neutral stochastic delay differential equation with Markov switching,we add the Poisson jump,which make the model much more difficult.By using Lyapunov function,nonnegative semi-martingale convergence theorem,Young inequality,basic inequality and Gronwall inequality,the existence uniqueness of global solution and the new criterion for the almost surely exponential stability of neutral stochastic delay differential equations with Markov switching and Poisson jump are established.