In this paper, we are concerned with convergence rate of Euler–Maruyama scheme for stochastic differential equations with Hölder–Dini continuous drifts. The key contributions are as follows: (i) by means of regularity of non-degenerate Kolmogrov equation, we investigate convergence rate of Euler–Maruyama scheme for a class of stochastic differential equations which allow the drifts to be Dini continuous and unbounded; (ii) by the aid of regularization properties of degenerate Kolmogrov equation, we discuss convergence rate of Euler–Maruyama scheme for a range of degenerate stochastic differential equations where the drifts are Hölder–Dini continuous of order \(\frac{2}{3}\) with respect to the first component and are merely Dini-continuous concerning the second component.

In this article, we investigate the strong convergence of the Euler–Maruyama method and stochastic theta method for stochastic differential delay equations with jumps. Under a global Lipschitz condition, we not only prove the strong convergence, but also obtain the rate of convergence. We show strong convergence under a local Lipschitz condition and a linear growth condition. Moreover, it is the first time that we obtain the rate of the strong convergence under a local Lipschitz condition and a linear growth condition, i.e., if the local Lipschitz constants for balls of radius R are supposed to grow not faster than log R.

The recent article [ 2 Higham , D.J. , Mao , X. , and Stuart , A.M. 2002 . Strong convergence of numerical methods for nonlinear stochastic differential equations . SIAM J. Num. Anal. 40 : 1041 – 1063 . [Crossref], [Web of Science ®] [Google Scholar] ] reveals the strong convergence of the Euler–Maruyama solution to the exact solution of a stochastic differential equation under the local Lipschitz condition. However, it does not provide us with an order of convergence. In this note, we will show the rate of convergence still under the local Lipschitz condition, but the local Lipschitz constants of the drift coefficient, valid on balls of radius R, are supposed not to grow faster than log R while those of the diffusion coefficient are not than .