用于有效风险估计的多层嵌套模拟
Multilevel nested simulation for efficient risk estimation
摘要：
我们调查的问题计算嵌套期望的形式$\mathbb{P}[\mathbb{E}[X|Y] \!\geq\!0]\!=\!\mathbb{E}[\textrm{H}(\mathbb{E}[X|Y])]$其中$\textrm{H}$是海维赛德函数。例如，当估计金融投资组合的巨大损失的概率时，就会出现这种嵌套的预期。我们提出了一种方法，该方法将使用多层蒙特卡罗（MLMC）进行嵌套期望的想法与自适应选择内部期望近似中的样本数量的想法相结合，如Broadie等人，2011年）。我们提出并分析了一种算法，自适应地选择每个MLMC水平上的内部样本的数量，并证明了所得到的MLMC方法与自适应采样具有$\mathcal{O}\left( \varepsilon^{-2}|\log\varepsilon|^2 \right)$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。我们还提出了一个随机根查找算法，结合我们的自适应方法，可用于计算其他风险措施，如风险价值（VaR）和条件风险价值（CVaR），后者是实现与$\mathcal{O}\left(\varepsilon^{-2}\right)$复杂性。
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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