In the field of computational finance, it is common for the quantity of interest to be expected values of functions of random variables via stochastic differential equations (SDEs). For SDEs with globally Lipschitz coefficients and commutative diffusion coefficients, the explicit Milstein scheme, relying on only Brownian increments and thus easily implementable, can be combined with the multilevel Monte Carlo (MLMC) method proposed by Giles \cite{giles2008multilevel} to give the optimal overall computational cost $\mathcal{O}(\epsilon^{-2})$, where $\epsilon$ is the required target accuracy. For multi-dimensional SDEs that do not satisfy the commutativity condition, a kind of one-half order truncated Milstein-type scheme without L\'evy areas is introduced by Giles and Szpruch \cite{giles2014antithetic}, which combined with the antithetic MLMC gives the optimal computational cost under globally Lipschitz conditions. In the present work, we turn to SDEs with non-globally Lipschitz continuous coefficients, for which a family of modified Milstein-type schemes without L\'evy areas is proposed. The expected one-half order of strong convergence is recovered in a non-globally Lipschitz setting, where the diffusion coefficients are allowed to grow superlinearly. This helps us to analyze the relevant variance of the multilevel estimator and the optimal computational cost is finally achieved for the antithetic MLMC. The analysis of both the convergence rate and the desired variance in the non-globally Lipschitz setting is highly non-trivial and non-standard arguments are developed to overcome some essential difficulties. Numerical experiments are provided to confirm the theoretical findings.

翻译：在计算金融领域，人们通常关注通过随机微分方程（SDEs）定义的随机变量函数的期望值。对于具有全局Lipschitz系数和交换扩散系数的SDEs，仅依赖布朗增量且易于实现的显式Milstein格式，可与Giles \cite{giles2008multilevel}提出的多层次蒙特卡罗（MLMC）方法结合，获得最优整体计算复杂度$\mathcal{O}(\epsilon^{-2})$，其中$\epsilon$为所需目标精度。对于不满足交换条件的高维SDEs，Giles和Szpruch \cite{giles2014antithetic}引入了一种不含Lévy区域的半阶截断Milstein型格式，该格式结合对偶MLMC在全局Lipschitz条件下实现了最优计算成本。本文进一步研究非全局Lipschitz连续系数的SDEs，并提出一系列不含Lévy区域的修正Milstein型格式。在扩散系数允许超线性增长的非全局Lipschitz设定下，恢复了预期的强收敛半阶精度。这有助于分析多层次估计量的相关方差，并最终在对偶MLMC中实现最优计算成本。非全局Lipschitz设定下收敛速率和期望方差的分析高度非平凡，本文发展了非标准论证以克服若干本质困难。数值实验验证了理论结果。