In the field of computational finance, one is commonly interested in the expected value of a financial derivative whose payoff depends on the solution of stochastic differential equations (SDEs). For multi-dimensional SDEs with non-commutative diffusion coefficients in the globally Lipschitz setting, a kind of one-half order truncated Milstein-type scheme without L\'evy areas was recently introduced by Giles and Szpruch (2014), which combined with the antithetic multilevel Monte Carlo (MLMC) gives the optimal overall computational cost $\mathcal{O}(\epsilon^{-2})$ for the required target accuracy $\epsilon$. Nevertheless, many nonlinear SDEs in applications have non-globally Lipschitz continuous coefficients and the corresponding theoretical guarantees for antithetic MLMC are absent in the literature. In the present work, we aim to fill the gap and analyze antithetic MLMC in a non-globally Lipschitz setting. First, we propose a family of modified Milstein-type schemes without L\'evy areas to approximate SDEs with non-globally Lipschitz continuous coefficients. The expected one-half order of strong convergence is recovered in a non-globally Lipschitz setting, where even the diffusion coefficients are allowed to grow superlinearly. This then helps us to analyze the relevant variance of the multilevel estimator and the optimal computational cost is finally achieved for the antithetic MLMC. Since getting rid of the L\'evy areas destroys the martingale properties of the scheme, the analysis of both the convergence rate and the desired variance becomes highly non-trivial in the non-globally Lipschitz setting. By introducing an auxiliary approximation process, we develop non-standard arguments to overcome the essential difficulties. Numerical experiments are provided to confirm the theoretical findings.

翻译：在计算金融领域，人们通常关注金融衍生品期望值的计算，其收益依赖于随机微分方程（SDE）的解。对于全局Lipschitz条件下具有非交换扩散系数的多维SDE，Giles与Szpruch（2014）近期提出了一种不含Lévy区域的半阶截断Milstein型格式，结合对立多级蒙特卡洛（MLMC）方法，可在目标精度$\epsilon$要求下实现$\mathcal{O}(\epsilon^{-2})$的最优总体计算成本。然而，实际应用中的许多非线性SDE具有非全局Lipschitz连续系数，而现有文献缺乏关于对立MLMC在此类情况下的理论保证。本研究旨在填补这一空白，分析非全局Lipschitz条件下的对立MLMC方法。首先，我们提出一族改进的不含Lévy区域的Milstein型格式，用于逼近具有非全局Lipschitz连续系数的SDE。在非全局Lipschitz条件下恢复了预期的半阶强收敛性，其中甚至允许扩散系数呈超线性增长。这有助于我们分析多级估计量的相关方差，最终实现对立MLMC的最优计算成本。由于消除Lévy区域破坏了格式的鞅性质，在非全局Lipschitz条件下分析收敛速率与目标方差变得尤为困难。通过引入辅助逼近过程，我们发展了非标准论证以克服本质性难点。数值实验验证了理论结果。