何志坚，男，1987年12月生，广东湛江人，华南理工大学数学学院副教授、硕士生导师。清华大学统计学博士，华南理工大学数学与应用数学本科。研究兴趣为统计计算与建模、随机模拟方法及其应用、金融工程。相关研究发表在统计学和计算科学领域顶级期刊，如统计学四大期刊Journal of the Royal Statistical Society: Series B，计算科学顶级期刊SIAM Journal on Numerical Analysis，SIAM Journal on Scientific Computing和Mathematics of Computation，运筹管理权威期刊European Journal of Operational Research等。博士论文获得新世界数学奖银奖。曾获第十四届金融系统工程与工程管理国际年会(FSERM2016)优秀论文奖。国家自然科学基金、广东省自然科学基金通讯评审专家。
统计学博士, 2015
清华大学
数学与应用数学学士, 2010
华南理工大学
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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 $\mathbb{R}^d$, 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 byproduct, we find that this rate also applies to randomized QMC integration with all terms of the analysis of variance decomposition of the discontinuous integrand, except the one of highest order.
This paper studies randomized quasi-Monte Carlo (QMC) sampling for discontinuous integrands having singularities along the boundary of the unit cube $ [0,1]^d$. Both discontinuities and singularities are extremely common in the pricing and hedging of financial derivatives and have a tremendous impact on the accuracy of QMC. It was previously known that the root mean square error of randomized QMC is only $ o(n^{1/2})$ for discontinuous functions with singularities. We find that under some mild conditions, randomized QMC yields an expected error of $ O(n^{-1/2-1/(4d-2)+\epsilon })$ for arbitrarily small $ \epsilon >0$. Moreover, one can get a better rate if the boundary of discontinuities is parallel to some coordinate axes. As a by-product, we find that the expected error rate attains $ O(n^{-1+\epsilon })$ if the discontinuities are QMC-friendly, in the sense that all the discontinuity boundaries are parallel to coordinate axes. The results can be used to assess the QMC accuracy for some typical problems from financial engineering.
We study the properties of points in $[0,1]^d$ generated by applying Hilbert’s space-filling curve to uniformly distributed points in $[0,1]$. For deterministic sampling we obtain a discrepancy of $O(n^{-1/d})$ for $d\ge2$. For random stratified sampling, and scrambled van der Corput points, we get a mean squared error of $O(n^{-1-2/d})$ for integration of Lipschitz continuous integrands, when $d\ge3$. These rates are the same as one gets by sampling on $d$ dimensional grids and they show a deterioration with increasing $d$. The rate for Lipschitz functions is however best possible at that level of smoothness and is better than plain IID sampling. Unlike grids, space-filling curve sampling provides points at any desired sample size, and the van der Corput version is extensible in $n$. We also introduce a class of piecewise Lipschitz functions whose discontinuities are in rectifiable sets described via Minkowski content. Although these functions may have infinite variation in the sense of Hardy and Krause, they can be integrated with a mean squared error of $O(n^{-1-1/d})$. It was previously known only that the rate was $o(n^{-1})$. Other space-filling curves, such as those due to Sierpinski and Peano, also attain these rates, while upper bounds for the Lebesgue curve are somewhat worse, as if the dimension were $\log_2(3)$ times as high.