英文标题：
《Ninomiya-Victoir scheme: strong convergence, antithetic version and
  application to multilevel estimators》
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作者：
Anis Al Gerbi, Benjamin Jourdain, Emmanuelle Cl\\\'ement
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最新提交年份：
2015
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英文摘要：
  In this paper, we are interested in the strong convergence properties of the Ninomiya-Victoir scheme which is known to exhibit weak convergence with order 2. We prove strong convergence with order $1/2$. This study is aimed at analysing the use of this scheme either at each level or only at the finest level of a multilevel Monte Carlo estimator: indeed, the variance of a multilevel Monte Carlo estimator is related to the strong error between the two schemes used on the coarse and fine grids at each level. Recently, Giles and Szpruch proposed a scheme permitting to construct a multilevel Monte Carlo estimator achieving the optimal complexity $O\\left(\\epsilon^{-2}\\right)$ for the precision $\\epsilon$. In the same spirit, we propose a modified Ninomiya-Victoir scheme, which may be strongly coupled with order $1$ to the Giles-Szpruch scheme at the finest level of a multilevel Monte Carlo estimator. Numerical experiments show that this choice improves the efficiency, since the order $2$ of weak convergence of the Ninomiya-Victoir scheme permits to reduce the number of discretization levels.
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中文摘要：
在本文中，我们对Ninomiya-Victoir格式的强收敛性感兴趣，该格式具有2阶弱收敛性。我们证明了订单为1/2美元时的强收敛性。在每一个层次上，分析两个多层蒙特卡罗方案的误差，实际上是在每一个多层蒙特卡罗方案中使用一个多层蒙特卡罗方案。最近，Giles和Szpruch提出了一个方案，允许构造一个多级蒙特卡罗估值器，以达到最优复杂度$O\\left（\\epsilon^{-2}\\right）$来获得精度$\\epsilon$。本着同样的精神，我们提出了一个改进的Ninomiya Victoir方案，该方案可能与Giles Szpruch方案在多层蒙特卡罗估计的最佳水平上的1美元订单强耦合。数值实验表明，这种选择提高了效率，因为Ninomiya Victoir格式的弱收敛阶为2$，可以减少离散级别的数量。
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分类信息：

一级分类：Quantitative Finance        数量金融学
二级分类：Computational Finance        计算金融学
分类描述：Computational methods, including Monte Carlo, PDE, lattice and other numerical methods with applications to financial modeling
计算方法，包括蒙特卡罗，偏微分方程，格子和其他数值方法，并应用于金融建模
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一级分类：Mathematics        数学
二级分类：Probability        概率
分类描述：Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory
概率论与随机过程的理论与应用：例如中心极限定理，大偏差，随机微分方程，统计力学模型，排队论
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