On High-Discrepancy Sequences

Kyushu University 21st Century COE Program Development of Dynamic Mathematics with High Functionality九州大学21世紀COEプログラム「機能数理学の構築と展開」First, it is pointed out that the uniform distribution of points in $[0, 1]^d$ is not always a necessary condition for every function in a proper subset of the class of all Riemann integrable functions to have the arithmetic mean of function values at the points converging to its integral over $[0, 1]^d$ as the number of points goes to infinity. We introduce a formal definition of the $d$-dimensional high-discrepancy sequences, which are not uniformly distributed in $[0, 1]^d$, and present motivation for the application of these sequences to high-dimensional numerical integration. Then, we prove that there exist non-uniform $(infty, d)$-sequences which provide the convergence rate $O(N^{−1})$ for the integration of a certain class of $d$-dimensional Walsh function series, where $N$ is the number of points

一様分布論とその応用

1.ディスクレパンシー 2.金融問題への応用例科学・技術の研究課題への数学アプローチ : 数学モデリングの基礎と展開基礎編第5部 : 応用数

ANALYSIS OF THE ANOMALY OF ran10 GENERATOR IN MONTE CARLO PRICING OF FINANCIAL DERIVATIVES

Abstract Recently, Paskov reported that the use of a certain pseudo-random number generator, rani(), given in Numerical Recipes in C, First Edition makes Monte Carlo simulations for pricing financial derivatives converge to wrong values. In this paper, we trace Paskov's experiment, investigate the characteristics and the generation algorithm of the pseudo-random number generator in question, and explain why the wrong convergences occur. We then present a method for avoiding such wrong convergences. A variance reduction procedure is applied, together with a method for obtaining more precise values, and its correctness is examined. We also investigate whether statistical tests for pseudo-random numbers can detect the cause of wrong convergences. 1

Exact Cubature for a Class of Functions of Maximum Effective Dimension

We consider high dimensional integration in a broad class of functions where all elements have maximum effective dimension. We show that there exists an exact cubature with only two points. Therefore, not only the convergence but also the worst case error of quasi-Monte Carlo need not depend on the effective dimension at all

A survey of high-discrepancy sequences

This article surveys recent results and open questions on high-discrepancy sequences.MI: Global COE Program Education-and-Research Hub for Mathematics-for-IndustryグローバルCOEプログラム「マス･フォア･インダストリ教育研究拠点