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.
Table of Contents
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. Kac–Rice 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 CIOŁEK 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 . 147
Benoite DE SAPORTA and Francois DUFOUR
5.1. Introduction 147
5.2. Simulation of piecewise-deterministic Markov processes 149
5.3. Optimal stopping 150
5.3.1. Assumptions and notations 150
5.3.2. Dynamic programming 153
5.3.3. Quantized approximation 154
5.4. Exit time 158
5.4.1. Problem setting and assumptions 158
5.4.2. Recursive formulation 159
5.4.3. Numerical approximation 161
5.5. Numerical example 162
5.5.1. Piecewise-deterministic Markov model 162
5.5.2. Deterministic time to reach the boundary 164
5.5.3. Quantization 166
5.5.4. Optimal stopping 167
5.5.5. Exit time 169
5.6. Conclusion 170
5.7. Bibliography 171
Chapter 6. Rupture Detection in Fatigue Crack Propagation 173
Romain AZAIS, Anne GEGOUT-PETIT and Florine GRECIET
6.1. Phenomenon of crack propagation 173
6.1.1. Virkler’s data 174
6.2. Modeling crack propagation 175
6.2.1. Deterministic models 175
6.2.2. Sources of uncertainties 177
6.2.3. Stochastic models 178
6.3. PDMP models of propagation 183
6.3.1. Relevance of PDMP models 183
6.3.2. Multiplicative model 185
6.3.3. One-jump models 186
6.4. Rupture detection 193
6.4.1. Length at versus time t . 193
6.4.2. Growth rate dat /dt versus ΔKt in log scale 194
6.5. Conclusion and perspectives 203
6.6. Bibliography 204
Chapter 7. Piecewise-deterministic Markov Processes for Spatio-temporal Population Dynamics . 209
Candy ABBOUD, Rachid SENOUSSI and Samuel SOUBEYRAND
7.1. Introduction 209
7.1.1. Models of population dynamics 209
7.1.2. Spatio-temporal PDMP for population dynamics 210
7.1.3. Chapter contents 212
7.2. Stratified dispersal models 212
7.2.1. Reaction–diffusion equations for modeling short-distance dispersal 212
7.2.2. Stratified diffusion 215
7.2.3. Coalescing colony model with Allee effect 216
7.2.4. A PDMP based on reaction–diffusion for modeling invasions with multiple introductions 221
7.3. Metapopulation epidemic model 223
7.3.1. Spatially realistic Levins model 223
7.3.2. A colonization PDMP 224
7.3.3. Bayesian inference approach 229
7.3.4. Markov chain Monte Carlo algorithm 235
7.3.5. Examples of results 236
7.4. Stochastic approaches for modeling spatial trajectories 237
7.4.1. Conditioning a Brownian motion by punctual observations 239
7.4.2. Movements with jumps 242
7.4.3. The Doléans–Dade exponential semi-martingales 247
7.4.4. Statistical issues 249
7.5. Conclusion 252
7.6. Bibliography 252
List of Authors 257
Index 259
