Many problems in finance lead to the numerical integration of sometimes extremely high-dimensional integrands. When applying low-discrepancy point sets (= QMC-methods) for the numerical evaluation of these integrals then classical error -analysis-tools like the Koksma-Hlawka inequality are not strong enough to give sharp enough error estimates. Nevertheless QMC-methods work very well in many cases, although this good performance until know cannot be justified on the basis of theoretical results. For example: When a Brownian Bridge algorithm is applied then it has turned out that many very high-dimensional problems can be attacked by QMC-methods.

Based on a weighted version of the Koksma-Hlawka inequality (given by Sloan, Wozniakowski and Wasilikowski) we try to give a theoretical and quantitative explanation of this (until now) just empirical fact.

Further we try to improve and to optimize the Brownian Bridge algorithm, which leads to problems in Diophantine approximation.

Please send comments and corrections to Thomas Klausner.