Convergence And Stability Analysis Of The Split-step ? Method For Neutral Stochastic Delay Differential Equations

Since the influence of environmental noise on system changes is considered,stochastic differential equations can more accurately describe the phenomena and the rules of the development of thing in real life than deterministic differential equations.Neutral stochastic delay differential equations(NSDDEs)are an important class of stochastic differential equations.NSDDEs depend not only on the current and past states,but also on rate of change over the past period of time.Moreover,NSDDEs are widely applied in biology,chemical,aerodynamics and engineering.Because most of the NSDDEs cannot be analytically solved,it is particularly important to study their numerical methods.The convergence and stability theory of numerical solutions is an important research topic in numerical analysis.This article discusses the stability and convergence of a class of split-step methods for NSDDEs.Firstly,the domestic and foreign research progress of the split-step method for NSDDEs and the preliminary knowledge required in the paper are introduced.Then,for NSDDEs,it is proven that the strong convergence order of the split-step method is 1/2 if the drift and diffusion coefficients satisfies the global Lipschitz condition with respect to the non-delay term,and they satisfies the polynomial growth condition with respect to the delay term,moreover,the neutral term stisfies the polynomial growth condition.Further,it is proven that the split step method is asymptotically and exponentially stable in mean square sense and under limited stepsizes if the drift and diffusion coefficients satisfy the local Lipschitz condition and the linear growth condition with respect to both the delay term and the non-delay term,the drift coefficient satisfies the one-sided Lipschitz condition,and the neutral term satisfies the compression condition.Finally,the correctness of the theoretical results is proved by numerical examples.