Mathematics and Visualization

By: Neil Dodgson (Editor), Michael S. Floater (Editor), Malcolm Sabin (Editor)

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Hardcover | 21 September 2004

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452 Pages

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23.5 x 15.88 x 1.91

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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.

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Non-Fiction Psychology Mathematics Geometry Computing & I.T.Graphical & Digital Media Applications 3D Graphics & Modelling Applied Mathematics Mathematical Modelling Computer Science Calculus & Mathematical Analysis Numerical Analysis

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Published: 22nd October 2010

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Introduction	p. 1
A Very Brief History of Financial Time Series	p. 1
Contents of the Monograph	p. 3
Parameter Estimation in a General Conditionally Heteroscedastic Time Series Model	p. 4
Whittle Estimation in GARCH(1,1)	p. 10
Structure of the Monograph	p. 11
Some Mathematical Tools	p. 13
Stationarity and Ergodicity	p. 13
Uniform Convergence via the Ergodic Theorem	p. 17
Bochner Expectation	p. 19
The Ergodic Theorem for Sequences of B-valued Random Elements	p. 22
Matrix Norms	p. 23
Weak Convergence in <$>{op C}<$>(K,<$>{op R}^{dprime}<$>)	p. 24
Exponentially Fast Almost Sure Convergence	p. 26
Stochastic Recurrence Equations	p. 29
Financial Time Series: Facts and Models	p. 37
Stylized Facts of Financial Log-return Data	p. 39
Uncorrelated Observations	p. 39
Time-varying Volatility (Conditional Heteroscedasticity)	p. 41
Heavy-tailed and Asymmetric Unconditional Distribution	p. 41
Leverage Effects	p. 43
ARMA Models	p. 44
Conditionally Heteroscedastic Time Series Models	p. 48
AG ARCH Models	p. 48
EGARCH Models	p. 60
Stochastic Volatility Models	p. 61
Parameter Estimation: An Overview	p. 63
Estimation for ARM A Processes	p. 63
Gaussian Quasi Maximum Likelihood Estimation	p. 63
Least-squares Estimation	p. 68
Whittle Estimation	p. 69
Estimation for GARCH Processes	p. 72
Quasi Maximum Likelihood Estimation	p. 73
Whittle Estimation	p. 79
Quasi Maximum Likelihood Estimation in ConditionallyHeteroscedastic Time Series Models: A StochasticRecurrence Equations Approach	p. 85
Overview	p. 85
Stationarity, Ergodicity and Invertibility	p. 87
Existence of a Stationary Solution	p. 88
Invertibility	p. 92
Definition of the Function h_t	p. 97
Consistency of the QMLE	p. 99
Examples: Consistency	p. 102
EGARCH	p. 102
AGARCH(p,q)	p. 105
The First and Second Derivatives of h_t and h_t	p. 110
Asymptotic Normality of the QMLE	p. 116
Examples: Asymptotic Normality	p. 120
AGARCH(p,q)	p. 120
EGARCH	p. 124
Non-Stationarities	p. 131
Fitting AGARCH(1,1) to the NYSE Composite Data	p. 131
A Simulation Study	p. 132
Maximum Likelihood Estimation in Conditionally Heteroscedastic Time Series Models	p. 141
Consistency of the MLE	p. 143
Main Result	p. 143
Consistency of the MLE with Respect to Student t Innovations	p. 146
Misspecification of the Innovations Density	p. 148
Inconsistency of the MLE	p. 148
Misspecfication of <$>{cal D}<$> in the GARCH(p,q) Model	p. 154
Asymptotic Normality of the MLE	p. 157
Asymptotic Normality of the MLE with Respect to Student t Innovations	p. 165
Quasi Maximum Likelihood Estimation in a Generalized Conditionally Heteroscedastic Time Series Model with Heavy-tailed Innovations	p. 169
Stable Limits of Infinite Variance Martingale Transforms	p. 170
Infinite Variance Stable Limits of the QMLE	p. 172
Limit Behavior of the QMLE in GARCH(p,q) with Heavy-tailed Innovations	p. 175
Verification of Strong Mixing with Geometric Rate of (Y_t) in GARCH(p,q)	p. 179
Whittle Estimation in a Heavy-tailed GARCH(1,1) Model	p. 187
Introduction	p. 187
Limit Theory for the Sample Autocovariance Function	p. 189
Main Results	p. 191
Excursion: Yule-Walker Estimation in ARCH(p)	p. 194
Proof of Theorem 8.3.1	p. 195
Proof of Theorem 8.3.2	p. 200
References	p. 215
Author Index	p. 221
Index	p. 225
Table of Contents provided by Publisher. All Rights Reserved.

Mathematics and Visualization

At a Glance

Hardcover

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