Correction: Mean-square convergence rates of implicit Milstein type methods for SDEs with non-Lipschitz coefficients

class of implicit Milstein type methods is introduced and analyzed in the present article for stochastic differential equations (SDEs) with non-globally Lipschitz drift and diffusion coefficients. By incorporating a pair of method parameters θ , η∈ [0, 1] into both the drift and diffusion parts, the new schemes are indeed a kind of drift-diffusion double implicit methods. Within a general framework, we offer upper mean-square error bounds for the proposed schemes, based on certain error terms only getting involved with the exact solution processes. Such error bounds help us to easily analyze mean-square convergence rates of the schemes, without relying on a priori high-order moment estimates of numerical approximations. Putting further globally polynomial growth condition, we successfully recover the expected mean-square convergence rate of order one for the considered schemes with θ∈ [12, 1], η∈ [0, 1] . Also, some of the proposed schemes are applied to solve three SDE models evolving in the positive domain (0, ∞ ) . More specifically, the particular drift-diffusion implicit Milstein method ( θ = η = 1 ) is utilized to approximate the Heston 3/2 -volatility model and the stochastic Lotka-Volterra competition model. The semi-implicit Milstein method ( θ =1, η = 0 ) is used to solve the Ait-Sahalia interest rate model. Thanks to the previously obtained error bounds, we reveal the optimal mean-square convergence rate of the positivity preserving schemes under more relaxed conditions, compared with existing relevant results in the literature. Numerical examples are also reported to confirm the previous findings.