A note on the rate of convergence of the Euler-Maruyama method for stochastic differential equations

The recent article [2] 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.