英文标题：
《Coupling Importance Sampling and Multilevel Monte Carlo using Sample
Average Approximation》
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作者：
Ahmed Kebaier (LAGA), J\\\'er\\^ome Lelong (DAO, MATHRISK)
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最新提交年份：
2017
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英文摘要：
In this work, we propose a smart idea to couple importance sampling and Multilevel Monte Carlo (MLMC). We advocate a per level approach with as many importance sampling parameters as the number of levels, which enables us to compute the different levels independently. The search for parameters is carried out using sample average approximation, which basically consists in applying deterministic optimisation techniques to a Monte Carlo approximation rather than resorting to stochastic approximation. Our innovative estimator leads to a robust and efficient procedure reducing both the discretization error (the bias) and the variance for a given computational effort. In the setting of discretized diffusions, we prove that our estimator satisfies a strong law of large numbers and a central limit theorem with optimal limiting variance, in the sense that this is the variance achieved by the best importance sampling measure (among the class of changes we consider), which is however non tractable. Finally, we illustrate the efficiency of our method on several numerical challenges coming from quantitative finance and show that it outperforms the standard MLMC estimator.
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中文摘要：
在这项工作中，我们提出了一个聪明的想法，耦合重要性抽样和多级蒙特卡罗（MLMC）。我们提倡每层方法，其重要性抽样参数与层数相同，这使我们能够独立计算不同的层。使用样本平均近似进行参数搜索，这基本上包括将确定性优化技术应用于蒙特卡罗近似，而不是诉诸随机近似。我们创新的估计器为给定的计算工作量提供了一个稳健而有效的过程，减少了离散化误差（偏差）和方差。在离散化扩散的设置中，我们证明了我们的估计满足一个强大的大数定律和一个具有最优极限方差的中心极限定理，在这个意义上，这是通过最佳重要性抽样度量（在我们考虑的变化类别中）获得的方差，但这是不可处理的。最后，我们在几个来自定量金融的数值挑战上说明了我们的方法的有效性，并表明它优于标准的MLMC估计。
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分类信息：

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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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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