Convergence And Stability Of Numerical Solutions For Two Classes Of Nonlinear Stochastic Time-Delay Systems

Stochastic time-delay systems are widely used in plenty of fields.It is difficult to express their analytical solutions.In practical applications,the numerical solutions are used.Therefore,it is very meaningful to analyze the convergence of numerical solutionsIn this paper,on the one hand,we investigate the strong convergence of the truncated Euler-Maruyama algorithm for stochastic differential delay equations with Poisson jumps of the form:dx(t)=f(x(t),x(t-?))dt+g(x(t),x(t-?))dB(t)+h(x(t-),x((t-?)-))dN(t).When the coefficients satisfy the polynomial growth condition and the generalized Khasminskii-type condition,the strong convergence rate of the numerical solution is givenOn the other hand,we investigate the strong convergence rate and long-time stability of the partially truncated Euler-Maruyama algorithm for neutral stochastic differential delay equations with Markovian switching of the form:d[x(t)-D(x(t-?),r(t))]=f(t,x(t),x(t-?),r(t))dt+g(t,x(t),x(t-?),r(t))dB(t).When the super-linear parts of the coefficients satisfy the polynomial growth condition and the generalized Khasminskii-type condition,based on the characteristics of the partially truncated Euler-Maruyama algorithm,the strong convergence rate and the almost sure exponential stability of the numerical solution are investigated.In this paper,some examples are given to show that the theory can cover many highly nonlinear stochastic differential delay equations with Poisson jumps and highly nonlinear and nonautonomous neutral stochastic differential delay equations with Markovian switching.