Almost Sure Stability With General Rate Of Neutral Stochastic Delayed Hybrid Equations With Lévy Noise

With the development of stochastic analysis,stochastic system has a wide application prospect in different fields of pure mathematics and applied mathematics,such as dynamic system,probability science,quantum mechanics and electrodynamics,etc.In the study of stochastic differential equations,on the one hand,Markov chains and diffusion systems with jumps are generally more suitable to describe the random faults,abrupt changes or sudden perturbations that occur in many physical systems.In many practical models,due to the change of internal and external environment,the system may switch between some finite states,and Markov chain is a very common model to simulate this kind of system state switching.On the other hand,there exist time delays in many fields such as mechanical engineering,chemistry and chemical engineering,life sciences,finance,etc.Time delays are often the cause of instability,so it is necessary to consider the action of time delays in systems.Among them,the neutral system is a class of time delay system which depends on past and present values but that involves derivatives with delays as well as the function itself;as a class of special stochastic delay systems,stochastic pantograph equations(SPEs)with unbounded delay have been investigated by many scholars.The stability theory of numerical solutions is one of the key problems in numerical analysis,and it has many important applications in real-world.As we know,the almost sure stability of neutral stochastic delayed hybrid equations with Lévy noise(NSDHEs-LN)has not been studied before.In this paper,we combine time delay with pantograph delay,and this paper focuses on the almost sure stability of neutral stochastic delayed hybrid equations with Lévy noise(NSDHEs-LN).First of all,under the local Lipschitz condition and non-linear growth condition,this paper gives a sufficient criteria for the existence and uniqueness of solutions to neutral stochastic delayed hybrid equations with Lévy noise.Then,a kind of ψ-function is introduced and the almost sure stability of neutral stochastic delayed hybrid equations with Lévy noise is investigated,including the exponential stability and the polynomial stability.Besides,this paper establishes a sufficient criteria on the general decay stability of neutral stochastic delayed hybrid equations with Lévy noise,and it gives the upper bound of decay rate.After that,according to the M-matrix theory,we impose some visualized conditions for the coefficients which are more convenient to check,and we establish the sufficient criteria on the almost sure ψ-stability of neutral stochastic delayed hybrid equations with Lévy noise.Especially,the coefficients of considered system can be allowed to be higher order nonlinear.Finally,two examples of numerical simulation are given to show the effectiveness of our results.