Asymptotic error distribution of the Euler method for SDEs with non-Lipschitz coefficients

In [Stochastic Analysis: 331–346, 1991, Annals of Probability 26: 267–307, 1998] Kurtz and Protter resp. Jacod and Protter specify the asymptotic error distribution of the Euler method for stochastic differential equations (SDEs) with smooth coefficients growing at most linearly. The required differentiability and linear growth of the coefficients rule out some popular SDEs as for instance the Cox–Ingersoll–Ross (CIR) model, the Heston model, or the stochastic Brusselator. In this article, we partially extend one of the fundamental results in [Jacod and Protter, Annals of Probability 26: 267–307, 1998], so that also the mentioned examples are covered. Moreover, we compare by means of simulations the asymptotic error distributions of the CIR model and the geometric Brownian motion with mean reversion.