The Partially Truncated Euler-Maruyama Methods For Two Class Of Super-linear Stochastic Delay Differential Equation

Recently,there are some results of the stochastic differential equations that have an exact solution under the global Lipschitz assumption or local Lipschitz condition plus the linear growth conditions.However,for most of the stochastic differential equations,the linear growth condition is too strict.When the linear growth condition is replaced by a more general Khasminskii condition,the existence and uniqueness of stochastic differential equations had been proved.As we know,the exact solution for most of the stochastic differential equations is difficult to obtain.Thus the numerical computation as an important tool to investigate the dynamics.For the nonlinear stochastic differential equation,the research shows that when the drift or diffusion coefficient is super-linear the classical EM method is diver-gence.So some articles modified the explicit EM such that this is applicable numerical methods to deal with the super-linear stochastic differential equations.such as tamed EM scheme,stopping EM scheme,truncated EM scheme.We,in this paper,will u-tile the partially truncated EM methods to deal with the two-class of stochastic delay differential equations:stochastic pantograph differential equation and stochastic delay differential equation with the Markovian switching.Stochastic pantograph differential equation,as a special stochastic delay differen-tial equation,has lots of care.There exist some results about the convergence and sta-bility for the linear stochastic pantograph equation.However,although the existence and uniqueness of the super-linear stochastic pantograph equations under the Khasminskii condition had been testified,the numerical result not investigated.Thus,in this paper,we mainly use the partially truncated Euler-Maruyama method to study the mean-square convergence of nonlinear stochastic pantograph differential equations under the Khas-minskii condition and estimate the mean square convergence order.Meanwhile,we will give the almost sure polynomial stability and mean-square polynomial stability.Lately,lots of articles have been studied the stochastic delay differential equations under linear growth conditions ever thought the Khasminskii assumption.But,the stochastic differential equations with the Markovian switching has more complexity.Therefore,we also use the partially truncated Euler-Maruyama method to deal with the stochastic delay differential equation with Markovian switching and variable delay to obtain the convergence order and almost sure exponent stability.