Statistical Inference for Piecewise-deterministic Markov Processes

Piecewise-deterministic Markov processes form a class of stochastic models with a sizeable scope of applications: biology, insurance, neuroscience, networks, finance... Such processes are defined by a deterministic motion punctuated by random jumps at random times, and offer simple yet challenging models to study. Nevertheless, the issue of statistical estimation of the parameters ruling the jump mechanism is far from trivial. Responding to new developments in the field as well as to current research interests and needs, Statistical inference for piecewise-deterministic Markov processes offers a detailed and comprehensive survey of state-of-the-art results. It covers a wide range of general processes as well as applied models. The present book also dwells on statistics in the context of Markov chains, since piecewise-deterministic Markov processes are characterized by an embedded Markov chain corresponding to the position of the process right after the jumps.

Auteur
AZAÏS Romain, Inria Nancy - Grand Est, Institut Elie Cartan de Lorraine. BOUGUET Florian, Inria Nancy - Grand Est, Institut Elie Cartan de Lorraine.

Contenu
Preface xi
Romain AZAÏS and Florian BOUGUET

List of Acronyms xiii

Introduction xv
Romain AZAÏS and Florian BOUGUET

Chapter 1. Statistical Analysis for Structured Models on Trees 1
*Marc HOFFMANN and Adelaide OLIVIER*

1.1. Introduction 1

1.1.1. Motivation 1

1.1.2. Genealogical versus temporal data 2

1.2. Size-dependent division rate 4

1.2.1. From partial differential equation to stochastic models 4

1.2.2. Non-parametric estimation: the Markov tree approach 6

1.2.3. Sketch of proof of Theorem 1.1 10

1.3. Estimating the age-dependent division rate 16

1.3.1. Heuristics and convergence of empirical measures 17

1.3.2. Estimation results 20

1.3.3. Sketch of proof of Theorem 1.4 24

1.4. Bibliography 37

Chapter 2. Regularity of the Invariant Measure and Non-parametric Estimation of the Jump Rate 39
*Pierre HODARA, Nathalie KRELL and Eva LOCHERBACH*

2.1. Introduction 39

2.2. Absolute continuity of the invariant measure 43

2.2.1. The dynamics 43

2.2.2. An associated Markov chain and its invariant measure 45

2.2.3. Smoothness of the invariant density of a single particle 47

2.2.4. Lebesgue density in dimension N 50

2.3. Estimation of the spiking rate in systems of interacting neurons 51

2.3.1. Harris recurrence 55

2.3.2. Properties of the estimator 56

2.3.3. Simulation results 58

2.4. Bibliography 61

Chapter 3. Level Crossings and Absorption of an Insurance Model 65
*Romain AZAÏS and Alexandre GENADOT*

3.1. An insurance model 65

3.2. Some results about the crossing and absorption features 70

3.2.1. Transition density of the post-jump locations 70

3.2.2. Absorption time and probability 71

3.2.3. KacRice formula 74

3.3. Inference for the absorption features of the process 77

3.3.1. Semi-parametric framework 77

3.3.2. Estimators and convergence results 79

3.3.3. Numerical illustration 81

3.4. Inference for the average number of crossings 89

3.4.1. Estimation procedures 89

3.4.2. Numerical application 90

3.5. Some additional proofs 92

3.5.1. Technical lemmas 92

3.5.2. Proof of Proposition 3.3 97

3.5.3. Proof of Corollary 3.2 98

3.5.4. Proof of Theorem 3.5 100

3.5.5. Proof of Theorem 3.6 102

3.5.6. Discussion on the condition (C2G) 103

3.6. Bibliography 104

Chapter 4. Robust Estimation for Markov Chains with Applications to Piecewise-deterministic Markov Processes 107
*Patrice BERTAIL, Gabriela CIOEK and Charles TILLIER*

4.1. Introduction 107

4.2. (Pseudo)-regenerative Markov chains 109

4.2.1. General Harris Markov chains and the splitting technique 110

4.2.2. Regenerative blocks for dominated families 111

4.2.3. Construction of regeneration blocks 112

4.3. Robust functional parameter estimation for Markov chains 114

4.3.1. The influence function on the torus 115

4.3.2. Example 1: sample means 116

4.3.3. Example 2: M-estimators 117

4.3.4. Example 3: quantiles 118

4.4. Central limit theorem for functionals of Markov chains and robustness 118

4.5. A Markov view for estimators in PDMPs 121

4.5.1. Example 1: Sparre Andersen model with barrier 122

4.5.2. Example 2: kinetic dietary exposure model 125

4.6. Robustness for risk PDMP models 127

4.6.1. Stationary measure 127

4.6.2. Ruin probability 132

4.6.3. Extremal index 136

4.6.4. Expected shortfall 138

4.7. Simulations 140

4.8. Bibliography 144 **Chapter 5. Numerical Method for Control of Piecewise-deterministic Markov Processes ...