Stability Analysis Of Stochastic Differential Equations Driven By Lévy Process

The stability problem of stochastic differential equations has been a hot topic in recent years and there have been many important and classic theories.However,it is not enough to describe many systems in the real world.In order to more accurately model some real systems,a new branch begins to appear,namely stochastic differential equations with random perturbations and noise.It is known that stochastic differential equations with Gaussian noise have been extensively studied and applied,but Gaussian noise is not suitable for simulating some large disturbances or jumps.Lévy noise compensates for this shortcoming,which can be used to describe random disturbances in many real-world systems.Therefore,this paper mainly studies several kinds of stability problems of stochastic differential equations with Lévy process.Based on stochastic analysis theory,the system uses the Lévy type stochastic integral,It(?) formula,Lyapunov function method,Dini derivative and some inequalities to obtain the system.The sufficient conditions for several types of stability,the conditions are simple and general.At the end of the article,the effectiveness and practicability of the results are shown by examples.The main research results of this paper are as follows:1.The domestic and foreign research status and existing problems of the stability of stochastic differential equations driven by Lévy process are analyzed and discussed.2.The definitions of the probability stability,the pth moment asymptotic stability,the pth moment exponential stability and the almost sure exponential stability of the stochastic differential equations driven by Lévy process are given,using the It(?) formula,tools such as Dini derivative and Kunita estimation give the sufficiency conditions for these types of stability.3.Based on the sufficiency conditions given in 2,the stability of the time-varying stochastic differential equations driven by Lévy process is discussed.Finally,the numerical results show that the results are valid and practical.