Stability Of Two Classes Of Stochastic Differential Equations

Stochastic systems are ubiquitous in many fields.The premise that a system can operate normally is that the system must be stable.Therefore,it is very important to study the stability of the system.Based on this,it is significant to study the stability of stochastic differential equations(SDEs)under the influence of different factors.Therefore,this paper mainly studies the stability of two different stochastic systems.The first system is devoted to study a class of nonlinear stochastic differential delay equations(SDDEs)with Poisson jump.In comparison to the Brownian motion,the jump leads to the discontinuity of sample paths,which makes the analysis more difficult In this section,we first introduce the local Lipschitz condition and a new nonlinear growth condition,which is weaker than those in the previous literature.Then by virtue of Lyapunov function and semi-martingale convergence theorem,we prove that the considered stochastic system has a unique global solution.Moreover,we also discuss the pth moment exponential stability and the almost surely exponential stability of solutions.Finally,an example is given to illustrate the effectiveness of theoretical results.In the other system,we analyze the stability of Impulsive stochastic functional differential equations(ISFDEs)with delayed impulses and Markovian switching.In this section,we introduced some new conditions to investigate the pth exponential stability and almost exponential stability of ISFDEs.Moreover,the coefficient of the estimated upper bound for the Lyapunov functions time-derivative has been improved to be either positive or negative.And some methods have been extended to the IS-DDEs with multiple time delays and linear ISDEs systems.The investigated topic is interesting and the mathematic explanation of the proposed solutions is also nice.Fur-thermore,the system is accompanied by an adequate set of experiments for evaluating the effectiveness of the solutions we proposed.To sum up,many factors of interference and uncertainty are taken into account in the system model in this paper,which makes the two types of random system model studied more general and the research method also has great innovation.