A 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 $\theta, \eta \in [0, 1]$ 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 $\theta \in [\tfrac12, 1], \eta \in [0, 1]$. Also, some of the proposed schemes are applied to solve three SDE models evolving in the positive domain $(0, \infty)$. More specifically, the particular drift-diffusion implicit Milstein method ($ \theta = \eta = 1 $) is utilized to approximate the Heston $\tfrac32$-volatility model and the stochastic Lotka-Volterra competition model. The semi-implicit Milstein method ($\theta =1, \eta = 0$) 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.

翻译：本文引入并分析了一类适用于漂移和扩散系数非全局Lipschitz的随机微分方程的隐式Milstein型方法。通过在漂移和扩散部分分别引入一对方法参数$\theta, \eta \in [0, 1]$，新格式实际上是一种漂移-扩散双重隐式方法。在一个一般性框架下，我们基于仅涉及精确解过程的特定误差项，给出了所提格式的均方误差上界。这种误差上界有助于我们在不依赖数值逼近先验高阶矩估计的情况下，简便地分析格式的均方收敛率。在进一步施加全局多项式增长条件后，我们成功恢复了所考虑格式在$\theta \in [\tfrac12, 1], \eta \in [0, 1]$时预期的一阶均方收敛率。此外，部分所提格式被应用于求解三个定义在正域$(0, \infty)$上的随机微分方程模型。具体而言，特定漂移-扩散隐式Milstein方法（$\theta = \eta = 1$）被用于逼近Heston $\tfrac32$-波动率模型和随机Lotka-Volterra竞争模型；半隐式Milstein方法（$\theta =1, \eta = 0$）被用于求解Ait-Sahalia利率模型。得益于先前获得的误差上界，与已有相关结果相比，我们在更宽松条件下揭示了保持正性格式的最优均方收敛率。文中还报告了数值算例以验证前述结论。