This book is a collective volume authored by leading scientists in the field of stochastic modelling, associated statistical topics and corresponding applications. The main classes of stochastic processes for dependent data investigated throughout this book are Markov, semi-Markov, autoregressive and piecewise deterministic Markov models. The material is divided into three parts corresponding to: (i) Markov and semi-Markov processes, (ii) autoregressive processes and (iii) techniques based on divergence measures and entropies. A special attention is payed to applications in reliability, survival analysis and related fields.…mehr
This book is a collective volume authored by leading scientists in the field of stochastic modelling, associated statistical topics and corresponding applications. The main classes of stochastic processes for dependent data investigated throughout this book are Markov, semi-Markov, autoregressive and piecewise deterministic Markov models. The material is divided into three parts corresponding to: (i) Markov and semi-Markov processes, (ii) autoregressive processes and (iii) techniques based on divergence measures and entropies. A special attention is payed to applications in reliability, survival analysis and related fields.
"Vlad Stefan BARBU1: 1Associate Professor of Mathematics (Statistics) - HDR (Habilitation to Conduct Research); Laboratory of Mathematics Raphaël Salem, University of Rouen - Normandy, France Nicolas VERGNE: Associate Professor of Mathematics (Statistics); Laboratory of Mathematics Raphaël Salem, University of Rouen - Normandy, France"
Inhaltsangabe
Preface xi Vlad Stefan BARBU and Nicolas VERGNE
Part 1. Markov and Semi-Markov Processes 1
Chapter 1. Variable Length Markov Chains, Persistent Random Walks: A Close Encounter 3 Peggy CÉNAC, Brigitte CHAUVIN, Frédéric PACCAUT and Nicolas POUYANNE
1.1. Introduction 3
1.2. VLMCs: definition of the model 6
1.3. Definition and behavior of PRWs 9
1.3.1. PRWs in dimension one 9
1.3.2. PRWs in dimension two 13
1.4. VLMC: existence of stationary probability measures 15
1.5. Where VLMC and PRW meet 19
1.5.1. Semi-Markov chains and Markov additive processes 19
1.5.2. PRWs induce semi-Markov chains 20
1.5.3. Semi-Markov chain of the alpha-LIS in a stable VLMC 22
1.5.4. The meeting point 23
1.6. References 27
Chapter 2. Bootstraps of Martingale-difference Arrays Under the Uniformly Integrable Entropy 29 Salim BOUZEBDA and Nikolaos LIMNIOS
2.1. Introduction and motivation 29
2.2. Some preliminaries and notation 30
2.3. Main results 35
2.4. Application for the semi-Markov kernel estimators 36
2.5. Proofs 41
2.6. References 45
Chapter 3. A Review of the Dividend Discount Model: From Deterministic to Stochastic Models 47 Guglielmo D'AMICO and Riccardo DE BLASIS
3.1. Introduction 47
3.2. General model 48
3.3. Gordon growth model and extensions 50
3.3.1. Gordon model 50
3.3.2. Two-stage model 51
3.3.3. H model 52
3.3.4. Three-stage model 52
3.3.5. N-stage model 53
3.3.6. Other extensions 53
3.4. Markov chain stock models 54
3.4.1. Hurley and Johnson model 54
3.4.2. Yao model 56
3.4.3. Markov stock model 57
3.4.4. Multivariate Markov chain stock model 61
3.5. Conclusion 64
3.6. References 65
Chapter 4. Estimation of Piecewise-deterministic Trajectories in a Quantum Optics Scenario 69 Romain AZAIS and Bruno LEGGIO
4.1. Introduction 69
4.1.1. The postulates of quantum mechanics 69
4.1.2. Dynamics of open quantum Markovian systems 71
4.1.3. Stochastic wave function: quantum dynamics as PDPs 74
4.1.4. Estimation for PDPs 76
4.2. Problem formulation 77
4.2.1. Atom-field interaction 77
4.2.2. Piecewise-deterministic trajectories 78
4.2.3. Measures 80
4.3. Estimation procedure 80
4.3.1. Strategy 80
4.3.2. Least-square estimators 82
4.3.3. Numerical experiments 83
4.4. Physical interpretation 86
4.5. Concluding remarks 87
4.6. References 88
Chapter 5. Identification of Patterns in a Semi-Markov Chain 91 Brenda Ivette GARCIA-MAYA and Nikolaos LIMNIOS
5.1. Introduction 91
5.2. The prefix chain 93
5.3. The semi-Markov setting 94
5.4. The hitting time of the pattern 100
5.5. A genomic application 102
5.6. Concluding remarks 106
5.7. References 106
Part 2. Autoregressive Processes 109
Chapter 6. Time Changes and Stationarity Issues for Continuous Time Autoregressive Processes of Order p 111 Valérie GIRARDIN and Rachid SENOUSSI
6.1. Introduction 111
6.2. Basics 112
6.3. Stationary AR processes 114
6.3.1. Formulas for the two first-order moments 114
6.3.2. Examples 116
6.3.3. Conditions for stationarity of CAR1(p) processes 118
Chapter 1. Variable Length Markov Chains, Persistent Random Walks: A Close Encounter 3 Peggy CÉNAC, Brigitte CHAUVIN, Frédéric PACCAUT and Nicolas POUYANNE
1.1. Introduction 3
1.2. VLMCs: definition of the model 6
1.3. Definition and behavior of PRWs 9
1.3.1. PRWs in dimension one 9
1.3.2. PRWs in dimension two 13
1.4. VLMC: existence of stationary probability measures 15
1.5. Where VLMC and PRW meet 19
1.5.1. Semi-Markov chains and Markov additive processes 19
1.5.2. PRWs induce semi-Markov chains 20
1.5.3. Semi-Markov chain of the alpha-LIS in a stable VLMC 22
1.5.4. The meeting point 23
1.6. References 27
Chapter 2. Bootstraps of Martingale-difference Arrays Under the Uniformly Integrable Entropy 29 Salim BOUZEBDA and Nikolaos LIMNIOS
2.1. Introduction and motivation 29
2.2. Some preliminaries and notation 30
2.3. Main results 35
2.4. Application for the semi-Markov kernel estimators 36
2.5. Proofs 41
2.6. References 45
Chapter 3. A Review of the Dividend Discount Model: From Deterministic to Stochastic Models 47 Guglielmo D'AMICO and Riccardo DE BLASIS
3.1. Introduction 47
3.2. General model 48
3.3. Gordon growth model and extensions 50
3.3.1. Gordon model 50
3.3.2. Two-stage model 51
3.3.3. H model 52
3.3.4. Three-stage model 52
3.3.5. N-stage model 53
3.3.6. Other extensions 53
3.4. Markov chain stock models 54
3.4.1. Hurley and Johnson model 54
3.4.2. Yao model 56
3.4.3. Markov stock model 57
3.4.4. Multivariate Markov chain stock model 61
3.5. Conclusion 64
3.6. References 65
Chapter 4. Estimation of Piecewise-deterministic Trajectories in a Quantum Optics Scenario 69 Romain AZAIS and Bruno LEGGIO
4.1. Introduction 69
4.1.1. The postulates of quantum mechanics 69
4.1.2. Dynamics of open quantum Markovian systems 71
4.1.3. Stochastic wave function: quantum dynamics as PDPs 74
4.1.4. Estimation for PDPs 76
4.2. Problem formulation 77
4.2.1. Atom-field interaction 77
4.2.2. Piecewise-deterministic trajectories 78
4.2.3. Measures 80
4.3. Estimation procedure 80
4.3.1. Strategy 80
4.3.2. Least-square estimators 82
4.3.3. Numerical experiments 83
4.4. Physical interpretation 86
4.5. Concluding remarks 87
4.6. References 88
Chapter 5. Identification of Patterns in a Semi-Markov Chain 91 Brenda Ivette GARCIA-MAYA and Nikolaos LIMNIOS
5.1. Introduction 91
5.2. The prefix chain 93
5.3. The semi-Markov setting 94
5.4. The hitting time of the pattern 100
5.5. A genomic application 102
5.6. Concluding remarks 106
5.7. References 106
Part 2. Autoregressive Processes 109
Chapter 6. Time Changes and Stationarity Issues for Continuous Time Autoregressive Processes of Order p 111 Valérie GIRARDIN and Rachid SENOUSSI
6.1. Introduction 111
6.2. Basics 112
6.3. Stationary AR processes 114
6.3.1. Formulas for the two first-order moments 114
6.3.2. Examples 116
6.3.3. Conditions for stationarity of CAR1(p) processes 118
6.4. Time transforms 125
6.4.1. Properties of time transforms 125
6.4
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