英文标题：
《Multilevel nested simulation for efficient risk estimation》
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作者：
Michael B. Giles, Abdul-Lateef Haji-Ali
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最新提交年份：
2019
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英文摘要：
  We investigate the problem of computing a nested expectation of the form $\\mathbb{P}[\\mathbb{E}[X|Y] \\!\\geq\\!0]\\!=\\!\\mathbb{E}[\\textrm{H}(\\mathbb{E}[X|Y])]$ where $\\textrm{H}$ is the Heaviside function. This nested expectation appears, for example, when estimating the probability of a large loss from a financial portfolio. We present a method that combines the idea of using Multilevel Monte Carlo (MLMC) for nested expectations with the idea of adaptively selecting the number of samples in the approximation of the inner expectation, as proposed by (Broadie et al., 2011). We propose and analyse an algorithm that adaptively selects the number of inner samples on each MLMC level and prove that the resulting MLMC method with adaptive sampling has an $\\mathcal{O}\\left( \\varepsilon^{-2}|\\log\\varepsilon|^2 \\right)$ complexity to achieve a root mean-squared error $\\varepsilon$. The theoretical analysis is verified by numerical experiments on a simple model problem. We also present a stochastic root-finding algorithm that, combined with our adaptive methods, can be used to compute other risk measures such as Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR), with the latter being achieved with $\\mathcal{O}\\left(\\varepsilon^{-2}\\right)$ complexity.
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中文摘要：
我们研究了$\\mathbb{P}[\\mathbb{E}[X | Y]\\形式的嵌套期望的计算问题！\\geq \\！0]\\!=\\!\\mathbb{E}[\\textrm{H}（\\mathbb{E}[X | Y]）]$，其中$\\textrm{H}$是Heaviside函数。例如，在估计金融投资组合出现巨额亏损的可能性时，就会出现这种嵌套预期。我们提出了一种方法，将多层蒙特卡罗（MLMC）用于嵌套期望的思想与（Broadie et al.，2011）提出的在近似内部期望的情况下自适应选择样本数的思想相结合。我们提出并分析了一种在每个MLMC层次上自适应选择内部样本数的算法，并证明了所得到的自适应采样MLMC方法具有$\\数学{O}\\ left（\\ varepsilon^{-2}| \\ log \\ varepsilon | ^ 2 \\ right）$复杂性，以实现均方根误差$\\ varepsilon$。通过一个简单模型问题的数值实验验证了理论分析的正确性。我们还提出了一种随机寻根算法，该算法与我们的自适应方法相结合，可用于计算其他风险度量，如风险值（VaR）和条件风险值（CVaR），后者是通过$\\数学{O}\\左（\\ varepsilon ^{-2}\\右）$复杂性实现的。
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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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