Improved Halton sequences and discrepancy bounds

For about fifteen years, the surprising success of quasi-Monte Carlo methods in finance has been raising questions that challenge our understanding of these methods. At the origin are numerical experiments performed with so-called GSobol’ and GFaure sequences by J. Traub and his team at Columbia University, following the pioneering work of S. Tezuka in 1993 on generalizations of Niederreiter (t, s)−sequences, especially with t = 0 (Faure sequences). Then in the early 2000, another breakthrough was achieved by E. Atanassov, who found clever generalizations of Halton sequences by means of permutations that are even asymptotically better than Niederreiter-Xing sequences in high dimensions. Unfortunately, detailed investigations of these GHalton sequences, together with numerical experiments, show that this good asymptotic behavior is obtained at the expense of remaining terms and is not sensitive to specific choices of permutations of Atanassov. As the theory fails, the reasons of the success of GHalton, as well as GFaure, must be sought elsewhere, for instance in specific selections of good scramblings by means of tailor-made permutations. In this paper, we report on our assertions above and we give some tracks to tentatively remove a part of the mystery of QMC success.