概率的自适应多层蒙特卡罗
Adaptive Multilevel Monte Carlo for Probabilities
摘要：
我们考虑$\mathbb{P}[G\in Ω]$的数值逼近，其中$d$维随机变量$G$不能直接采样，但有一个层次的越来越准确的近似$\{G_\ell\}_{\ell\in\mathbb{N}}$可以采样。标准蒙特卡罗估计的成本与此设置中的准确性不成比例，因为它使近似和采样成本复合。多层蒙特卡罗的直接应用稍微改进了这种成本缩放，但是返回次优的计算复杂性，因为概率的估计涉及$G_\ell$的不连续泛函。我们提出了一个通用的自适应框架，这是能够返回MLMC复杂性光滑或Lipschitz泛函的$G_\ell$。我们的假设和数值分析保持一般允许的方法被用于广泛的一类问题。我们提出了嵌套模拟的风险估计，其中$G = \mathbb{E}[X|Y]$近似的内部蒙特卡罗估计的数值实验。进一步的实验给出了数字期权定价，涉及一个近似的$d$维的价格。
英文摘要：
We consider the numerical approximation of $\mathbb{P}[G\in Ω]$ where the $d$-dimensional random variable $G$ cannot be sampled directly, but there is a hierarchy of increasingly accurate approximations $\{G_\ell\}_{\ell\in\mathbb{N}}$ which can be sampled. The cost of standard Monte Carlo estimation scales poorly with accuracy in this setup since it compounds the approximation and sampling cost. A direct application of Multilevel Monte Carlo improves this cost scaling slightly, but returns sub-optimal computational complexities since estimation of the probability involves a discontinuous functional of $G_\ell$. We propose a general adaptive framework which is able to return the MLMC complexities seen for smooth or Lipschitz functionals of $G_\ell$. Our assumptions and numerical analysis are kept general allowing the methods to be used for a wide class of problems. We present numerical experiments on nested simulation for risk estimation, where $G = \mathbb{E}[X|Y]$ is approximated by an inner Monte Carlo estimate. Further experiments are given for digital option pricing, involving an approximation of a $d$-dimensional SDE.
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