Uniform Distribution and Quasi-Monte Carlo Methods (eBook)

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Peter Kritzer, Harald Niederreiter, Friedrich Pillichshammer, and Arne Winterhof, JKU Linz, Austria.

Preface 5
Contents 7
Metric number theory, lacunary series and systems of dilated functions 11
1 Uniform distribution modulo 1 12
2 Metric number theory 14
3 Discrepancy 16
4 Lacunary series 17
5 Almost everywhere convergence 20
6 Sums involving greatest common divisors 22
Strong uniformity 27
1 Introduction 27
2 Superuniformity and super-duper uniformity 36
2.1 Superuniformity of the typical billiard paths 36
2.2 Super-duper uniformity of the 2-dimensional ray 47
3 Superuniformmotions 51
3.1 Billiards in other shapes 51
3.2 Superuniformity of the geodesics on an equifacial tetrahedron surface 52
Discrepancy theory and harmonic analysis 55
1 Introduction 55
2 Exponential sums 56
3 Fourier analysis methods 59
3.1 Rotated rectangles 59
3.2 The lower bound for circles 61
3.3 Further remarks 63
4 Dyadic harmonic analysis: discrepancy function estimates 64
4.1 Lp -discrepancy, 1 < p <
65 
4.2 The L8 
66 
4.3 The other endpoint, L1 
68 
Explicit constructions of point sets and sequences with low discrepancy 73
1 Introduction 73
2 Lower bounds 75
3 Upper bounds 77
4 Digital nets and sequences 79
5 Walsh series expansion of the discrepancy function 81
6 The construction of finite point sets according to Chen and Skriganov 87
7 The construction of infinite sequences according to Dick and Pillichshammer 89
8 Extensions to the Lq 
92 
9 Extensions to Orlicz norms of the discrepancy function 93
Subsequences of automatic sequences and uniform distribution 97
1 Introduction 97
2 Automatic sequences 100
3 Subsequences along the sequence nc 
103 
4 Polynomial subsequences 105
5 Subsequences along the primes 108
On Atanassov’s methods for discrepancy bounds of low-discrepancy sequences 115
1 Introduction 115
2 Atanassov’s methods for Halton sequences 117
2.1 Review of Halton sequences 117
2.2 Review of previous bounds for the discrepancy of Halton sequences 118
2.3 Atanassov’s methods applied to Halton sequences 118
2.4 Scrambling Halton sequences with matrices 123
3 Atanassov’s method for 
128 
3.1 Review of (t,s)-sequences 
128 
3.2 Review of bounds for the discrepancy of (t,s)-sequences 
129 
3.3 Atanassov’smethod applied to (t,s)- 
129 
3.4 The special case of even bases for (t,s)-sequences 131
4 Atanassov’s methods for generalized Niederreiter sequences and (??, e, ??)- sequences 134
The hybrid spectral test: a unifying concept 137
1 Introduction 137
2 Adding digit vectors 139
3 Notation 142
4 The hybrid spectral test 144
5 Examples 147
5.1 Example I: Integration lattices 147
5.2 Example II: Extreme and star discrepancy 150
Tractability of multivariate analytic problems 157
1 Introduction 157
2 Tractability 159
3 A weighted Korobov space of analytic functions 164
4 Integration in H(Ks,a,b) 166
5 L2-approximation in 
172 
6 Conclusion and outlook 179
Discrepancy estimates for sequences: new results and open problems 181
1 Introduction 181
2 Metrical and average type discrepancy estimates for digital point sets and sequences and for good lattice point sets 184
3 Discrepancy estimates for and applications of hybrid sequences 191
4 Miscellaneous problems 195
A short introduction to quasi-Monte Carlo option pricing 201
1 Overview 201
2 Foundations of financial mathematics 202
2.1 Bonds, stocks and derivatives 202
2.2 Arbitrage and the no-arbitrage principle 204
2.3 The Black–Scholesmodel 206
2.4 SDE models 207
2.5 Lévy models 209
2.6 Examples 210
3 MC and QMC simulation 211
3.1 Nonuniform random number generation 211
3.2 Generation of Brownian paths 218
3.3 Generation of Lévy paths 224
3.4 Multilevel (quasi-)Monte Carlo 226
3.5 Examples 228
The construction of good lattice rules and polynomial lattice rules 233
1 Lattice rules and polynomial lattice rules 233
1.1 Lattice rules 234
1.2 Polynomial lattice rules 235
2 The worst-case error 237
2.1 Koksma–Hlawka error bound 237
2.2 Lattice rules 239
2.3 Polynomial lattice rules 242
3 Weighted worst-case errors 246
4 Some standard spaces 248
4.1 Lattice rules and Fourier spaces 248
4.2 Randomly-shifted lattice rules and the unanchored Sobolev space 249
4.3 Tent-transformed lattice rules and the cosine space 251
4.4 Polynomial lattice rules and Walsh spaces 253
5 Component-by-component constructions 255
5.1 Component-by-component construction 255
5.2 Fast component-by-component construction 259
6 Conclusion 262
Index 267