This paper mainly investigates the strong convergence and stability of the truncated Euler-Maruyama (EM) method for stochastic differential delay equations with variable delay whose coefficients can be growing super-linearly. By constructing appropriate truncated functions to control the super-linear growth of the original coefficients, we present a type of the truncated EM method for such SDDEs with variable delay, which is proposed to be approximated by the value taken at the nearest grid points on the left of the delayed argument. The strong convergence result (without order) of the method is established under the local Lipschitz plus generalized Khasminskii-type conditions and the optimal strong convergence order $1/2$ can be obtained if the global monotonicity with U function and polynomial growth conditions are added to the assumptions. Moreover, the partially truncated EM method is proved to preserve the mean-square and H_\infty stabilities of the true solutions. Compared with the known results on the truncated EM method for SDDEs, a better order of strong convergence is obtained under more relaxing conditions on the coefficients, and more refined technical estimates are developed so as to overcome the challenges arising due to variable delay. Lastly, some numerical examples are utilized to confirm the effectiveness of the theoretical results.
翻译:本文主要调查了微缩的Euler-Maruyama(EM)方法的强烈趋同性和稳定性,该方法的延迟差价差价方程式具有可变延迟率,其系数可能增长超线性。我们通过建立适当的缺线功能来控制原系数的超线性增长,为这种SDDEs提供了一种微缩的EM方法,其延迟率可变多。建议用延迟参数左侧最近的网格点的数值来比较。该方法的强烈趋同结果(不按顺序)是在当地Lipschitz加上普遍的Khasminskii型条件和最佳强势趋同单价1/2美元下确立的。如果将全球与U函数的单一性增长和多线性增长条件加到假设之外,我们就可以达到最佳的1/2美元。此外,部分减速的EM方法已被证明可以维持真实解决办法的中值和Hüinfty稳定性稳定值。与SDDEs的松散式EM方法的已知结果相比(不按顺序),在SDDEs的平流化方法上,一个较精确的精确的理论趋近的趋同率是更好的排序。