Abstract This paper focuses mainly on the Euler scheme of stochastic delay differential equations with locally Lipschitz coefficients. The convergence in probability of the Euler scheme and the corresponding weak limit process of the normalized error process are derived. Furthermore, this paper also considers a class of specific degenerate stochastic delay equations and obtains the associated weak limit process for the stronger error process. The error parameter of this stronger error process for such a degenerate system is n instead of n in the normalized error process. This causes substantial challenges in the analysis and proofs and the weak limit process also becomes more complicated and involves some additional terms. This result is new and interesting even for the non-delay case.
Liu et al. (Mon,) studied this question.