英文标题：
《Ninomiya-Victoir scheme: strong convergence, antithetic version and
application to multilevel estimators》
---
作者：
Anis Al Gerbi, Benjamin Jourdain, Emmanuelle Cl\\\'ement
---
最新提交年份：
2015
---
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.
---
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。本研究旨在分析该方案在每个层次上的使用情况，或仅在多层蒙特卡罗估计量的最精细层次上的使用情况：事实上，多层蒙特卡罗估计量的方差与在每个层次的粗网格和细网格上使用的两个方案之间的强误差有关。最近，Giles和Szpruch提出了一个方案，该方案允许构造一个多级蒙特卡罗估计量，以达到精度$\\ε$\\的最优复杂性$\\左（\\ε^{-2}\\右）$。本着同样的精神，我们提出了一种改进的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格式的弱收敛阶数为2$，可以减少离散化层数。
---
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
--
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
--

---
PDF下载：
-->