We consider the numerical approximation of $\mathbb{P}[G\in \Omega]$ 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.
翻译:我们考虑的是美元(mathbb{P})[G\in\in\Omega]的数值近似值,因为美元(d)维度随机随机变量(G$)无法直接抽样,但有一个越来越准确的近似值($ ⁇ G ⁇ ell ⁇ ell\ell\in\mathb{N ⁇ }N ⁇ )的等级,可以抽样。标准蒙特卡洛的估算尺度成本与这一设置的准确性差强人意,因为它增加了近似值和取样成本。多级别蒙特卡洛的直接应用使这一成本比例略有提高,但返回了亚最佳计算复杂性,因为对概率的估计涉及不连续的功能$G ⁇ ell。我们提出了一个总体适应性框架,它能够返回光滑或利普施茨功能($)所观察到的MLMC复杂度。我们的假设和数字分析是一般性的,能够将方法用于广泛的问题类别。我们对风险估算的嵌套式模拟进行了数字实验,其中美元=\mathb{E}[X}[X]Y]美元是内部蒙特卡洛估计的近似值。进一步实验用于数字选项定价,涉及SDE的近似值。