Gerhard Larcher: On the Tractability of the Brownian Bridge Algorithm

This is a joint work with G. Leobacher and K. Scheicher.

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.

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