A weighted multilevel Monte Carlo method

arXiv - QuantFin - Computational Finance Pub Date : 2024-05-06 DOI:arxiv-2405.03453

Yu Li, Antony Ware

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Abstract

The Multilevel Monte Carlo (MLMC) method has been applied successfully in a wide range of settings since its first introduction by Giles (2008). When using only two levels, the method can be viewed as a kind of control-variate approach to reduce variance, as earlier proposed by Kebaier (2005). We introduce a generalization of the MLMC formulation by extending this control variate approach to any number of levels and deriving a recursive formula for computing the weights associated with the control variates and the optimal numbers of samples at the various levels. We also show how the generalisation can also be applied to the \emph{multi-index} MLMC method of Haji-Ali, Nobile, Tempone (2015), at the cost of solving a $(2^d-1)$-dimensional minimisation problem at each node when $d$ index dimensions are used. The comparative performance of the weighted MLMC method is illustrated in a range of numerical settings. While the addition of weights does not change the \emph{asymptotic} complexity of the method, the results show that significant efficiency improvements over the standard MLMC formulation are possible, particularly when the coarse level approximations are poorly correlated.

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加权多级蒙特卡罗方法

多层次蒙特卡洛（MLMC）方法自 Giles（2008 年）首次提出以来，已成功应用于多种场合。当只使用两个层次时，该方法可被视为一种控制变量方法来减少方差，正如 Kebaier（2005 年）早先提出的那样。我们介绍了 MLMC 方法的一般化，将这种控制变量方法扩展到任意数量的层次，并推导出一个递归公式，用于计算与控制变量相关的权重和各层次的最优样本数。我们还展示了如何将这一概括应用于 Haji-i- Haji-i- Haji-i- Haji-i- Haji-i- Haji-i-{多指数}MLMC 方法。Haji-Ali、Nobile、Tempone（2015）的 MLMC 方法，当使用 $d$ 指数维度时，每个节点都需要解决 $(2^d-1)$ 维度的最小化问题。在一系列数值设置中，加权 MLMC 方法的性能比较得到了说明。虽然增加权重并不会改变方法的渐近复杂度，但结果表明，与标准 MLMC 方法相比，效率有可能得到显著提高，尤其是当粗级近似相关性较差时。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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arXiv - QuantFin - Computational Finance

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