Advances in multiresolution for geometric modelling

Multiresolution methods in geometric modelling are concerned with the generation, representation, and manipulation of geometric objects at several levels of detail. Applications include fast visualization and rendering as well as coding, compression, and digital transmission of 3D geometric objects.

This book marks the culmination of the four-year EU-funded research project, Multiresolution in Geometric Modelling (MINGLE). The book contains seven survey papers, providing a detailed overview of recent advances in the various fields within multiresolution modelling, and sixteen additional research papers. Each of the seven parts of the book starts with a survey paper, followed by the associated research papers in that area. All papers were originally presented at the MINGLE 2003 workshop held at Emmanuel College, Cambridge, UK, 9-11 September 2003.

Author Notes

Neil Dodgson took his BSc in Computer Science and Physics at Massey University in New Zealand (1988) and his PhD in image processing at the University of Cambridge (1992). He is a Senior Lecturer in the Computer Laboratory at the University of Cambridge and is co-leader of the Rainbow Research Group. He has over fifty refereed publications in the areas of modelling for 3D computer graphics, human-figure animation, 3D displays, and image processing.

Malcolm Sabin worked on representation of aircraft shapes at British Aircraft Corporation in the late 1960s, there developing one of the earliest modern surface systems. He has been active in CAD, CAM and CAE ever since, especially in the field of surface representations and in subdivision in particular. He is employed by his own company, Numerical Geometry Ltd. which sells his time as a consultant, and maintains close contact with the Computer Laboratory and the Department of Applied Mathematics at the University of Cambridge.

1 Introduction	p. 1
1.1 A Very Brief History of Financial Time Series	p. 1
1.2 Contents of the Monograph	p. 3
1.2.1 Parameter Estimation in a General Conditionally Heteroscedastic Time Series Model	p. 4
1.2.2 Whittle Estimation in GARCH(1,1)	p. 10
1.3 Structure of the Monograph	p. 11
2 Some Mathematical Tools	p. 13
2.1 Stationarity and Ergodicity	p. 13
2.2 Uniform Convergence via the Ergodic Theorem	p. 17
2.2.1 Bochner Expectation	p. 19
2.2.2 The Ergodic Theorem for Sequences of B-valued Random Elements	p. 22
2.3 Matrix Norms	p. 23
2.4 Weak Convergence in {{\op C}} (K, {{\op R}}^{{d\prime}} )	p. 24
2.5 Exponentially Fast Almost Sure Convergence	p. 26
2.6 Stochastic Recurrence Equations	p. 29
3 Financial Time Series: Facts and Models	p. 37
3.1 Stylized Facts of Financial Log-return Data	p. 39
3.1.1 Uncorrelated Observations	p. 39
3.1.2 Time-varying Volatility (Conditional Heteroscedasticity)	p. 41
3.1.3 Heavy-tailed and Asymmetric Unconditional Distribution	p. 41
3.1.4 Leverage Effects	p. 43
3.2 ARMA Models	p. 44
3.3 Conditionally Heteroscedastic Time Series Models	p. 48
3.3.1 AG ARCH Models	p. 48
3.3.2 EGARCH Models	p. 60
3.4 Stochastic Volatility Models	p. 61
4 Parameter Estimation: An Overview	p. 63
4.1 Estimation for ARM A Processes	p. 63
4.1.1 Gaussian Quasi Maximum Likelihood Estimation	p. 63
4.1.2 Least-squares Estimation	p. 68
4.1.3 Whittle Estimation	p. 69
4.2 Estimation for GARCH Processes	p. 72
4.2.1 Quasi Maximum Likelihood Estimation	p. 73
4.2.2 Whittle Estimation	p. 79
5 Quasi Maximum Likelihood Estimation in ConditionallyHeteroscedastic Time Series Models: A StochasticRecurrence Equations Approach	p. 85
5.1 Overview	p. 85
5.2 Stationarity, Ergodicity and Invertibility	p. 87
5.2.1 Existence of a Stationary Solution	p. 88
5.2.2 Invertibility	p. 92
5.2.3 Definition of the Function h t	p. 97
5.3 Consistency of the QMLE	p. 99
5.4 Examples: Consistency	p. 102
5.4.1 EGARCH	p. 102
5.4.2 AGARCH(p,q)	p. 105
5.5 The First and Second Derivatives of h t and h t	p. 110
5.6 Asymptotic Normality of the QMLE	p. 116
5.7 Examples: Asymptotic Normality	p. 120
5.7.1 AGARCH(p,q)	p. 120
5.7.2 EGARCH	p. 124
5.8 Non-Stationarities	p. 131
5.9 Fitting AGARCH(1,1) to the NYSE Composite Data	p. 131
5.10 A Simulation Study	p. 132
6 Maximum Likelihood Estimation in Conditionally Heteroscedastic Time Series Models	p. 141
6.1 Consistency of the MLE	p. 143
6.1.1 Main Result	p. 143
6.1.2 Consistency of the MLE with Respect to Student t ¿ Innovations	p. 146
6.2 Misspecification of the Innovations Density	p. 148
6.2.1 Inconsistency of the MLE	p. 148
6.2.2 Misspecfication of {{\cal D}} in the GARCH(p,q) Model	p. 154
6.3 Asymptotic Normality of the MLE	p. 157
6.4 Asymptotic Normality of the MLE with Respect to Student t ¿ Innovations	p. 165
7 Quasi Maximum Likelihood Estimation in a Generalized Conditionally Heteroscedastic Time Series Model with Heavy-tailed Innovations	p. 169
7.1 Stable Limits of Infinite Variance Martingale Transforms	p. 170
7.2 Infinite Variance Stable Limits of the QMLE	p. 172
7.3 Limit Behavior of the QMLE in GARCH(p,q) with Heavy-tailed Innovations	p. 175
7.4 Verification of Strong Mixing with Geometric Rate of (Y t ) in GARCH(p,q)	p. 179
8 Whittle Estimation in a Heavy-tailed GARCH(1,1) Model	p. 187
8.1 Introduction	p. 187
8.2 Limit Theory for the Sample Autocovariance Function	p. 189
8.3 Main Results	p. 191
8.4 Excursion: Yule-Walker Estimation in ARCH(p)	p. 194
8.5 Proof of Theorem 8.3.1	p. 195
8.6 Proof of Theorem 8.3.2	p. 200
References	p. 215
Author Index	p. 221
Index	p. 225

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