“人大数学时间I”第四十六期：王小群
为促进学科交流融合、拓宽师生学术视野、释放科研创新活力，助力人大数学学科走向一流，91黑料 设立“人大数学时间”，以专题研讨、高端学术论坛为载体，搭建数学思想充分碰撞、优秀人才不断涌流、创造活力竞相迸发的舞台。“人大数学时间”将持之以恒，久久为功，立志通过交流与创新、提出重大问题，引领数学学科及相关领域的创新与发展，成为对我国数学发展有贡献意义的平台。以下为“人大数学时间I”第四十六期信息：
议程：
4月22日（星期三）15:30 王小群教授报告及前沿问题探讨
地点：立德楼701
线上：腾讯会议：825-501-374
题目：High-Dimensional Computation in Mathematical Finance
主讲专家：王小群，清华大学数学科学系
摘要：
High-dimensional integration problems are ubiquitous in finance and statistics. Their integrands are often unbounded, and the dimension can be as high as hundreds or thousands, or even infinite. Quasi–Monte Carlo (QMC) methods, which rely on deterministic low-discrepancy point sets, have proven powerful for such problems. Nevertheless, the classical Koksma–Hlawka inequality is not directly applicable for error analysis in this context, because the integrands typically possess unbounded variation. In this talk, we first review recent progress in explaining the “unreasonable effectiveness” of QMC methods for high‑dimensional financial problems, with an emphasis on the theory of weighted function spaces and effective dimension. We show how the curse of dimensionality can be broken while achieving optimal convergence rates by using QMC methods and by appropriately introducing weights to characterize the relative importance of different variables. Then, we propose a novel framework to analyze the convergence rate of (randomized) QMC methods for smooth unbounded integrands. Our approach is based on a projection technique that circumvents the singularities caused by unboundedness. Remarkably, we demonstrate that an even higher convergence rate can be attained by coupling a carefully designed importance sampling scheme with a randomized QMC rule using scrambled digital nets. Numerical experiments are provided to support the theoretical findings.
报告人简介：
王小群，清华大学数学科学系长聘教授，国家杰出青年科学基金获得者，国家高层次人才。研究领域为金融数学与计算金融学、蒙特卡洛与拟蒙特卡洛方法、数据科学和统计机器学习，以及高维分析与计算复杂性理论。在金融资产定价和金融风险管理、高维积分计算和降维方法、拟蒙特卡洛(Quasi-Monte Carlo)方法和计算复杂性，以及可信机器学习方面进行了系统研究，取得一系列创新性研究成果。学术成果发表在Management Science, Operations Research, SIAM Journal on Numerical Analysis, SIAM Journal on Scientific Computing, Mathematics of Computation, Numerische Mathematik, Journal of Complexity, Quantitative Finance等。
