Uniform Distribution and Quasi-Monte Carlo Methods

1 - Preface [Seite 5]
2 - Contents [Seite 7]
3 - Metric number theory, lacunary series and systems of dilated functions [Seite 11]
3.1 - 1 Uniform distribution modulo 1 [Seite 12]
3.2 - 2 Metric number theory [Seite 14]
3.3 - 3 Discrepancy [Seite 16]
3.4 - 4 Lacunary series [Seite 17]
3.5 - 5 Almost everywhere convergence [Seite 20]
3.6 - 6 Sums involving greatest common divisors [Seite 22]
4 - Strong uniformity [Seite 27]
4.1 - 1 Introduction [Seite 27]
4.2 - 2 Superuniformity and super-duper uniformity [Seite 36]
4.2.1 - 2.1 Superuniformity of the typical billiard paths [Seite 36]
4.2.2 - 2.2 Super-duper uniformity of the 2-dimensional ray [Seite 47]
4.3 - 3 Superuniformmotions [Seite 51]
4.3.1 - 3.1 Billiards in other shapes [Seite 51]
4.3.2 - 3.2 Superuniformity of the geodesics on an equifacial tetrahedron surface [Seite 52]
5 - Discrepancy theory and harmonic analysis [Seite 55]
5.1 - 1 Introduction [Seite 55]
5.2 - 2 Exponential sums [Seite 56]
5.3 - 3 Fourier analysis methods [Seite 59]
5.3.1 - 3.1 Rotated rectangles [Seite 59]
5.3.2 - 3.2 The lower bound for circles [Seite 61]
5.3.3 - 3.3 Further remarks [Seite 63]
5.4 - 4 Dyadic harmonic analysis: discrepancy function estimates [Seite 64]
5.4.1 - 4.1 Lp -discrepancy, 1 < p < [Seite 4.1 Lp -discrepancy, 1 < p < ]
8 - 65 [Seite 65]
5.4.2 - 4.2 The L8 [Seite 4.2 The L8]
discrepancy estimates - 66 [Seite 66]
5.4.3 - 4.3 The other endpoint, L1 [Seite 4.3 The other endpoint, L1]
- 68 [Seite 68]
6 - Explicit constructions of point sets and sequences with low discrepancy [Seite 73]
6.1 - 1 Introduction [Seite 73]
6.2 - 2 Lower bounds [Seite 75]
6.3 - 3 Upper bounds [Seite 77]
6.4 - 4 Digital nets and sequences [Seite 79]
6.5 - 5 Walsh series expansion of the discrepancy function [Seite 81]
6.6 - 6 The construction of finite point sets according to Chen and Skriganov [Seite 87]
6.7 - 7 The construction of infinite sequences according to Dick and Pillichshammer [Seite 89]
6.8 - 8 Extensions to the Lq [Seite 8 Extensions to the Lq]
discrepancy - 92 [Seite 92]
6.9 - 9 Extensions to Orlicz norms of the discrepancy function [Seite 93]
7 - Subsequences of automatic sequences and uniform distribution [Seite 97]
7.1 - 1 Introduction [Seite 97]
7.2 - 2 Automatic sequences [Seite 100]
7.3 - 3 Subsequences along the sequence nc [Seite 3 Subsequences along the sequence nc]
- 103 [Seite 103]
7.4 - 4 Polynomial subsequences [Seite 105]
7.5 - 5 Subsequences along the primes [Seite 108]
8 - On Atanassov's methods for discrepancy bounds of low-discrepancy sequences [Seite 115]
8.1 - 1 Introduction [Seite 115]
8.2 - 2 Atanassov's methods for Halton sequences [Seite 117]
8.2.1 - 2.1 Review of Halton sequences [Seite 117]
8.2.2 - 2.2 Review of previous bounds for the discrepancy of Halton sequences [Seite 118]
8.2.3 - 2.3 Atanassov's methods applied to Halton sequences [Seite 118]
8.2.4 - 2.4 Scrambling Halton sequences with matrices [Seite 123]
8.3 - 3 Atanassov's method for [Seite 3 Atanassov's method for]
(t,s)-sequences - 128 [Seite 128]
8.3.1 - 3.1 Review of (t,s)-sequences [Seite 3.1 Review of (t,s)-sequences]
- 128 [Seite 128]
8.3.2 - 3.2 Review of bounds for the discrepancy of (t,s)-sequences [Seite 3.2 Review of bounds for the discrepancy of (t,s)-sequences]
- 129 [Seite 129]
8.3.3 - 3.3 Atanassov'smethod applied to (t,s)- [Seite 3.3 Atanassov'smethod applied to (t,s)-]
sequences - 129 [Seite 129]
8.3.4 - 3.4 The special case of even bases for (t,s)-sequences [Seite 131]
8.4 - 4 Atanassov's methods for generalized Niederreiter sequences and (??, e, ??)- sequences [Seite 134]
9 - The hybrid spectral test: a unifying concept [Seite 137]
9.1 - 1 Introduction [Seite 137]
9.2 - 2 Adding digit vectors [Seite 139]
9.3 - 3 Notation [Seite 142]
9.4 - 4 The hybrid spectral test [Seite 144]
9.5 - 5 Examples [Seite 147]
9.5.1 - 5.1 Example I: Integration lattices [Seite 147]
9.5.2 - 5.2 Example II: Extreme and star discrepancy [Seite 150]
10 - Tractability of multivariate analytic problems [Seite 157]
10.1 - 1 Introduction [Seite 157]
10.2 - 2 Tractability [Seite 159]
10.3 - 3 A weighted Korobov space of analytic functions [Seite 164]
10.4 - 4 Integration in H(Ks,a,b) [Seite 166]
10.5 - 5 L2-approximation in [Seite 5 L2-approximation in]
H(Ks,a,b) - 172 [Seite 172]
10.6 - 6 Conclusion and outlook [Seite 179]
11 - Discrepancy estimates for sequences: new results and open problems [Seite 181]
11.1 - 1 Introduction [Seite 181]
11.2 - 2 Metrical and average type discrepancy estimates for digital point sets and sequences and for good lattice point sets [Seite 184]
11.3 - 3 Discrepancy estimates for and applications of hybrid sequences [Seite 191]
11.4 - 4 Miscellaneous problems [Seite 195]
12 - A short introduction to quasi-Monte Carlo option pricing [Seite 201]
12.1 - 1 Overview [Seite 201]
12.2 - 2 Foundations of financial mathematics [Seite 202]
12.2.1 - 2.1 Bonds, stocks and derivatives [Seite 202]
12.2.2 - 2.2 Arbitrage and the no-arbitrage principle [Seite 204]
12.2.3 - 2.3 The Black-Scholesmodel [Seite 206]
12.2.4 - 2.4 SDE models [Seite 207]
12.2.5 - 2.5 Lévy models [Seite 209]
12.2.6 - 2.6 Examples [Seite 210]
12.3 - 3 MC and QMC simulation [Seite 211]
12.3.1 - 3.1 Nonuniform random number generation [Seite 211]
12.3.2 - 3.2 Generation of Brownian paths [Seite 218]
12.3.3 - 3.3 Generation of Lévy paths [Seite 224]
12.3.4 - 3.4 Multilevel (quasi-)Monte Carlo [Seite 226]
12.3.5 - 3.5 Examples [Seite 228]
13 - The construction of good lattice rules and polynomial lattice rules [Seite 233]
13.1 - 1 Lattice rules and polynomial lattice rules [Seite 233]
13.1.1 - 1.1 Lattice rules [Seite 234]
13.1.2 - 1.2 Polynomial lattice rules [Seite 235]
13.2 - 2 The worst-case error [Seite 237]
13.2.1 - 2.1 Koksma-Hlawka error bound [Seite 237]
13.2.2 - 2.2 Lattice rules [Seite 239]
13.2.3 - 2.3 Polynomial lattice rules [Seite 242]
13.3 - 3 Weighted worst-case errors [Seite 246]
13.4 - 4 Some standard spaces [Seite 248]
13.4.1 - 4.1 Lattice rules and Fourier spaces [Seite 248]
13.4.2 - 4.2 Randomly-shifted lattice rules and the unanchored Sobolev space [Seite 249]
13.4.3 - 4.3 Tent-transformed lattice rules and the cosine space [Seite 251]
13.4.4 - 4.4 Polynomial lattice rules and Walsh spaces [Seite 253]
13.5 - 5 Component-by-component constructions [Seite 255]
13.5.1 - 5.1 Component-by-component construction [Seite 255]
13.5.2 - 5.2 Fast component-by-component construction [Seite 259]
13.6 - 6 Conclusion [Seite 262]
14 - Index [Seite 267]