On the Error Rate of Conditional Quasi-Monte Carlo for Discontinuous Functions
On the Error Rate of Conditional Quasi-Monte Carlo for Discontinuous Functions
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DOI：
10.1137/18m118270x
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发表时间：
2017-08
期刊：
影响因子：
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通讯作者：
Zhijian He
中科院分区：
文献类型：
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作者：
Zhijian He
本文研究了条件拟蒙特卡罗算法的收敛速度，它是条件蒙特卡罗算法的一种对应物。我们关注的是定义在整体上的不连续积分，它可以是无界的。在适当的条件下，我们证明了条件QMC不仅具有平滑效果（达到无限倍可微），而且与普通QMC相比，积分误差可以降低几个数量级。特别地，对于期权定价和希腊估计中的一些典型问题，对于任意小的$\epsilon> $，使用$n$样本的条件随机QMC产生的平均误差为$O(n^{-1+\epsilon})$。作为一个副产品，我们发现这个速率也适用于随机QMC积分与所有项的不连续被积函数的ANOVA分解，除了最高阶的一个。
This paper studies the rate of convergence for conditional quasi-Monte Carlo (QMC), which is a counterpart of conditional Monte Carlo. We focus on discontinuous integrands defined on the whole of $R^d$, which can be unbounded. Under suitable conditions, we show that conditional QMC not only has the smoothing effect (up to infinitely times differentiable), but also can bring orders of magnitude reduction in integration error compared to plain QMC. Particularly, for some typical problems in options pricing and Greeks estimation, conditional randomized QMC that uses $n$ samples yields a mean error of $O(n^{-1+\epsilon})$ for arbitrarily small $\epsilon>0$. As a by-product, we find that this rate also applies to randomized QMC integration with all terms of the ANOVA decomposition of the discontinuous integrand, except the one of highest order.