Study On Existence And Stability For Some Stochastic Differential Equations And Its Applications

Along with the development of science and technology, the demand for precision of describing system is getting higher and higher. The systems in real world are often disturbed by some uncertain factors, therefore it is difficult to describe these systems by using determined differential equation. For stochastic differential equation to modeling can be more authentic, accurate depiction of the status of the systems, thus in recent years stochastic differential equation has been widely applied in natural science, engi-neering technology, etc. Moreover, it is important to study the existence-uniqueness and stability for equation. Therefore, it is a very meaningful work to investigate the existence-uniqueness and stability of stochastic differential equation.In the thesis, the existence-uniqueness and stability of several stochastic differential equations are discussed. At first, the prosperities of solution for two class of stochastic differential equation driven by Levy process are investigated. Then, the stability of impulsive stochastic differential equation and stochastic switched system are discussed. The compendious frame and description of the thesis are given as follows:(1) The existence, uniqueness and continuity of the adapted solutions for neutral stochastic delay Volterra equations with singular kernels are discussed. It is noticed that delay phenomenon is widely existed in real life, and many results of the existence and uniqueness for the stochastic delay differential equation are obtained. However, There are little results of the existence and uniqueness for the stochastic delay Volterra equation. In this thesis, the existence-uniqueness and continuity of the adapted solu-tions for neutral stochastic delay Volterra equations with singular kernels are discussed. In addition, continuous dependence on the initial date is also investigated. Finally, s-tochastic Volterra equation with the kernel of fractional Brownian motion is studied to illustrate the effectiveness of our results.(2) The numerical solutions of doubly perturbed stochastic delay differential equa-tions driven by Levy process is considered. In the existing results, the existence and uniqueness of the solution for these equations is investigated. However, It is difficult to get an explicit solutions and hence require numerical solutions to estimate the prop- erties of the solution. By using the Euler-Maruyama method, we define the numerical solutions, and show that the numerical solutions converge to the true solutions under the local Lipschitz condition. Finally, we obtain the order of convergence under the global Lipschtiz condition.(3) The stability of stochastic delay differential systems with delayed impulses by Razumikhin methods was discussed. In the existing results, there are some stability results of impulsive stochastic differential delay equation are obtained. However, time delay are inevitably occurred in the transmission of the impulsive information, when the delay exists in stochastic differential equation. Hence, impulsive input delays should be considered. Some criteria on the pth moment exponential stability are obtained. It is shown that if a stochastic delay differential system is exponentially stable, then under some conditions, its stability is robust or weaken with respect to delayed impulses. Moreover, it is also shown that an unstable stochastic delay system can be successfully stabilized by delayed impulses. Finally, the effectiveness of the proposed results is illustrated by three examples.(4) By using the average interval of impulsive, the stability of stochastic delay differential systems is investigated. In most existing results, the constraints of the maximum impulsive interval or minimum impulsive interval are required. However, it is reasonable to see that high-density impulses happen in a certain interval and low-density impulses in other interval. Thus, it is conservatism to restrict the maximum or minimum impulsive interval. By using the average impulsive interval, the impulsive occurrence satisfies some condition, then the stochastic differential equation is expo-nential stability. Finally, two examples are provided to illustrate the effectiveness of the obtained results.(5) By using the average dwell time, the stability of nonlinear stochastic switched systems with delay is concerned. It is shown that when the Lyapunov like functions are decreasing in all active subsystems, the switched system is pth moment exponentially stable. Moreover, it is also shown that under some conditions the system can be pth moment stable and global asymptotically stable, where the Lyapunov like functions are increasing on some intervals. The effectiveness of the proposed results is illustrated by two examples.