Probabilistic Analysis of Multivariate GARCH Models (PDF)

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Inhaltsverzeichnis:Table of Contents:
Introductionii
1.Preliminaries1
1.1Markov Chains1
1.1.1Strict Stationarity and Stationarity2
1.1.2Invariant Measures3
1.1.3Irreducibility, Small Sets and Aperiodic Chains4
1.1.4Petite Sets5
1.1.5Feller Chains6
1.1.6Transience, Recurrence and Harris Recurrence8
1.1.7Ergodicity9
1.1.8ß ¿ Mixing9
1.1.9Criterion for Ergodicity and ß ¿ Mixing10
1.2Algebraic Geometry12
1.2.1Semi-algebraic and Algebraic Sets12
1.2.2Regular Points and Dimension of Algebraic Varieties15
1.2.3Regular Maps17
2.Autoregressive Processes defined by a Composition of a Regular Map and a Diffeomorphism20
2.1Introduction20
2.2Properties of the Image Measure21
2.3Semi-polynomial Markov Chains24
2.3.1Model and Assumptions25
2.3.2Algebraic Variety of States26
2.3.3Harris Recurrence, Ergodicity and ß ¿ Mixing28
3.Multivariate GARCH Models32
3.1Introduction and Notations32
3.2The vec and BEKK Models33
3.3Stationarity of Multivariate GARCH Models36
3.3.1Autoregressive Representation36
3.3.2Some Results from Linear Algebra38
3.3.3Verification of Assumption (A2)44
3.3.4Verification of Assumption (A3)46
3.3.5Foster - Lyapounov Condition (FL)49
3.3.6Harris Recurrence, Ergodicity and ß ¿ Mixing52
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