Table of Contents

Cover

Preface

Introduction

I.1. Piecewise-deterministic Markov processes
I.2. Connection with other stochastic processes
I.3. Book contents
I.4. Bibliography

1 Statistical Analysis for Structured Models on Trees

1.1. Introduction
1.2. Size-dependent division rate
1.3. Estimating the age-dependent division rate
1.4. Bibliography

2 Regularity of the Invariant Measure and Non-parametric Estimation of the Jump Rate

2.1. Introduction
2.2. Absolute continuity of the invariant measure
2.3. Estimation of the spiking rate in systems of interacting neurons
2.4. Bibliography

3 Level Crossings and Absorption of an Insurance Model

3.1. An insurance model
3.2. Some results about the crossing and absorption features
3.3. Inference for the absorption features of the process
3.4. Inference for the average number of crossings
3.5. Some additional proofs
3.6. Bibliography

4 Robust Estimation for Markov Chains with Applications to Piecewise-deterministic Markov Processes

4.1. Introduction
4.2. (Pseudo)-regenerative Markov chains
4.3. Robust functional parameter estimation for Markov chains
4.4. Central limit theorem for functionals of Markov chains and robustness
4.5. A Markov view for estimators in PDMPs
4.6. Robustness for risk PDMP models
4.7. Simulations
4.8. Bibliography

5 Numerical Method for Control of Piecewise-deterministic Markov Processes

5.1. Introduction
5.2. Simulation of piecewise-deterministic Markov processes
5.3. Optimal stopping
5.4. Exit time
5.5. Numerical example
5.6. Conclusion
5.7. Bibliography

6 Rupture Detection in Fatigue Crack Propagation

6.1. Phenomenon of crack propagation
6.2. Modeling crack propagation
6.3. PDMP models of propagation
6.4. Rupture detection
6.5. Conclusion and perspectives
6.6. Bibliography

7 Piecewise-deterministic Markov Processes for Spatio-temporal Population Dynamics

7.1. Introduction
7.2. Stratified dispersal models
7.3. Metapopulation epidemic model
7.4. Stochastic approaches for modeling spatial trajectories
7.5. Conclusion
7.6. Bibliography

List of Authors

Index

End User License Agreement

List of Tables

5 Numerical Method for Control of Piecewise-deterministic Markov Processes

Table 5.1. Numerical values of the parameters of the corrosion model
Table 5.2. Numerical results for the calculation of the value function for the corrosion process
Table 5.3. Approximation results for the distribution of the exit time.

6 Rupture Detection in Fatigue Crack Propagation

Table 6.1. Notations
Table 6.2. Statistics concerning the crack length at transition, the transition times in terms of number of cycles and the corresponding stress intensity factor range
Table 6.3. Normalized distance D and crack position at the end of the propagation d₁₆₀ as a function of the type of crack
Table 6.4. Statistics concerning the crack at transition, the transition times in terms of the number of cycles and the corresponding stress intensity factor range

List of Illustrations

Introduction

Figure I.1. Typical path of a growth-fragmentation process with τ(x) = τ
Figure I.2. Typical path of a pharmacokinetics process with τ(x) = τx
Figure I.3. Model of gene expression with a two-dimensional process
Figure I.4. Typical path of a telegraph process (blue: m = 1; red: m = −1). For a color version of this figure, see www.iste.co.uk/azais/markov.zip
Figure I.5. Typical path of an exponential zig-zag process (d = 3; blue: m = 1; red: m = 2; brown: m = 3). For a color version of this figure, see www.iste.co.uk/azais/markov.zip
Figure I.6. System of switching differential equations (d = 2; blue: m = 0; red: m = 1). For a color version of this figure, see www.iste.co.uk/azais/markov.zip

1 Statistical Analysis for Structured Models on Trees

Figure 1.1. Genealogical tree observed up to T = 7 for an age-dependent division rate B(a) = a² (60 cells). Blue: Ů_T; red: ∂U_T. For a color version of this figure, see www.iste.co.uk/azais/markov.zip
Figure 1.2. The same outcome organized at a genealogical level. For a color version of this figure, see www.iste.co.uk/azais/markov.zip

2 Regularity of the Invariant Measure and Non-parametric Estimation of the Jump Rate

Figure 2.1. Trajectory of five neurons. For a color version of this figure, see www.iste.co.uk/azais/markov.zip
Figure 2.2. Estimation of the intensity function λ(x) = x. For a color version of this figure, see www.iste.co.uk/azais/markov.zip
Figure 2.3. Estimation of the intensity function λ(x) = log(x + 1). For a color version of this figure, see www.iste.co.uk/azais/markov.zip
Figure 2.4. Estimation of the intensity function λ(x) = exp(x) – 1. For a color version of this figure, see www.iste.co.uk/azais/markov.zip

3 Level Crossings and Absorption of an Insurance Model

Figure 3.1. Two trajectories for the growth-fragmentation model starting from X₀ = 1.1 and whose characteristics are λ = r = 1 and G(u) = 11u¹⁰, as in the numerical example investigated in section 3.3.3. One of them is absorbed at the first jump time, whereas the other seems to escape the trapping set
Figure 3.2. Reference curve R(·, 2) and its estimates from the observation of n = 50, 75 or 100 random loss events (top left) with a zoom around R(2, 2) (top right), and the pointwise error on 100 replicates between R(·, 2) and (bottom). For a color version of this figure, see www.iste.co.uk/azais/markov.zip
Figure 3.3. Reference curve R(2, ·) and its estimates from the observation of n = 50, 75 or 100 random loss events (top left) with a zoom around R(2, 2) (top right), and the pointwise error on 100 replicates between R(2, ·) and (bottom). For a color version of this figure, see www.iste.co.uk/azais/markov.zip
Figure 3.4. Integrated square error on 100 replicates between R(·, 2) and its estimate (left) and between R(2, ·) and its estimate (right), from the observation of n = 50, 75 or 100 random loss events
Figure 3.5. Absorption probability p (approximated by p_m) and its estimates from the observation of n = 50, 75 or 100 random loss events and for m = 10 iterations of the estimated kernel (top), and the associated pointwise error on 100 replicates from n = 100 random loss events (bottom). For a color version of this figure, see www.iste.co.uk/azais/markov.zip
Figure 3.6. Integrated square error on 100 replicates between the absorption probability p (approximated by p_m) and its estimates from the observation of n = 50, 75 or 100 random loss events and for m = 10 iterations of the estimated kernel
Figure 3.7. Integrated square error on 100 replicates between t_m and its estimate from the observation of n = 50, 75 or 100 random loss events and for m = 1 (top left), m = 2 (top right), m = 3 (bottom left) and m = 4 (bottom right)
Figure 3.8. Distribution of the hitting time of Γ t_m(x) for x = 1.1 and m = 1, …, 6 and its estimates from the observation of n = 50, 75 or 100 random loss events. For a color version of this figure, see www.iste.co.uk/azais/markov.zip
Figure 3.9. Boxplots of the average number of continuous crossings for 100 replications. For each figure, the MC estimator is at the right of the KR estimator. Top: the level is x = 1.1 and the time step is increasing from left to right h = 0.01, 0.1, 0.2. Bottom: the level is x = 1.15 and the time step is increasing from left to right h = 0.01, 0.1, 0.2
Figure 3.10. Boxplots of the average number of continuous crossings for 100 replications. For each figure, the MC estimator is at the right of the KR estimator. Top: the level is x = 1.05 and the time step is increasing from left to right h = 0.01, 0.1, 0.2. Bottom: the level is x = 1.2 and the time step is increasing from left to right h = 0.01, 0.1, 0.2

4 Robust Estimation for Markov Chains with Applications to Piecewise-deterministic Markov Processes

Figure 4.1. Trajectory of the SA model with a barrier. The horizontal red line corresponds to a dividend barrier. For a color version of this figure, see www.iste.co.uk/azais/markov.zip
Figure 4.2. Trajectory of the KDEM process
Figure 4.3. Illustration of the splitting technique on the CL model with a barrier. Observations between two vertical green lines correspond to independent blocks. The barrier is the horizontal blue line, here at d = 3. The process starts at X(0) = 5. For a color version of this figure, see www.iste.co.uk/azais/markov.zip
Figure 4.4. Comparison between the true distribution (red) and the plug-in estimator (black) when there is no contamination. For a color version of this figure, see www.iste.co.uk/azais/markov.zip
Figure 4.5. Trajectory of the CL model when contamination is added (around 75) with a Dirac measure. For a color version of this figure, see www.iste.co.uk/azais/markov.zip
Figure 4.6. Comparison of the true distribution (red), the plug-in estimator (black) and the robustified estimator (green) when there is contamination. For a color version of this figure, see www.iste.co.uk/azais/markov.zip

5 Numerical Method for Control of Piecewise-deterministic Markov Processes

Figure 5.1. Simulated trajectories of the thickness loss over time for the corrosion process. For a color version of this figure, see www.iste.co.uk/azais/markov.zip
Figure 5.2. Quantization grids (2000 points) for the corrosion process: inter arrival times (abscissa) and thickness loss at the jump times (ordinate)
Figure 5.3. Reward function for the maintenance of the corrosion process
Figure 5.4. Examples of stopped trajectories for the corrosion process. For a color version of this figure, see www.iste.co.uk/azais/markov.zip
Figure 5.5. Histograms of approximate and optimal stopping times for the corrosion process (10⁵ Monte Carlo simulations)
Figure 5.6. Survival function of the service time of the corrosion process obtained through Monte Carlo simulations (dashed) and quantized approximation (solid), and the error with 2000 points in the quantization grids. For a color version of this figure, see www.iste.co.uk/azais/markov.zip

6 Rupture Detection in Fatigue Crack Propagation

Figure 6.1. Schematic illustration of the different regimes of FCP. The vertical dashed lines indicate the transition between crack propagation regimes
Figure 6.2. The 68 experimental crack length curves versus the number of cycles provided by [VIR 79]
Figure 6.3. Crack growth rate in terms of ΔK_t computed from the 68 experiments conducted in [VIR 79]
Figure 6.4. Simulation of crack propagation curves with only one jump and two modes before and after the jump
Figure 6.5. Experimental (solid line) and theoretical (dashed line) curves for the worst fitted propagation length curve (left) and for the best one (right). For a color version of this figure, see www.iste.co.uk/azais/markov.zip
Figure 6.6. End of propagation for three different experimental cracks and fitting by regime-switching models with Paris’ and Forman’s law for the second regime. For a color version of this figure, see www.iste.co.uk/azais/markov.zip
Figure 6.7. Top: first step of actualization method. Bottom: second step. Experimental measurements are drawn with dots, and continuous lines indicate the four (r = 4) nearest theoretical curves from the ten (p = 10) in total
Figure 6.8. Three experimental curves with the extreme curves of the prediction bundle. From top to bottom, rapid crack with D = 0 and D = 0.2 and slow crack with D = 11.6
Figure 6.9. Example of one curve of the simulated model.
Figure 6.10. Probability of Z according to X. For a color version of this figure, see www.iste.co.uk/azais/markov.zip
Figure 6.11. Example of one curve of the simulated model
Figure 6.12. Experimental (points) and theoretical (line) curves for the worst fitted propagation curve (left) and for the best one (right) in the da_t/dt versus ΔK_t representation. Vertical lines show the time of transition between regimes II and III. For a color version of this figure, see www.iste.co.uk/azais/markov.zip
Figure 6.13. Experimental (solid line) and theoretical (dashed line) curves for the worst fitted propagation curve (left) and for the best one (right) in a_t versus t. For a color version of this figure, see www.iste.co.uk/azais/markov.zip
Figure 6.14. Comparison of the times of transition estimations obtained by the EM algorithm in piecewise polynomial regression (abscissa) and fitting switching ODEs (ordinate)
Figure 6.15. Estimated jump rate of a crack according to its length.

7 Piecewise-deterministic Markov Processes for Spatio-temporal Population Dynamics

Figure 7.1. Illustrations of the flows and jumps of the coalescing colony model (left), the metapopulation epidemic model (center) and the simple velocity-jump model (right)
Figure 7.2. Unidimensional random walk model.
Figure 7.3. Numerical solution u(t, x) of Skellam model (equation [7.2]) in a bi-dimensional space (where x = (x, y)) with Neumann boundary conditions, at time 0 (top left), 3 (top right), 6 (bottom left) and 12 (bottom right). The dispersal coefficient and the intrinsic growth rate were fixed at (D, r) = (5 × 10⁻³, 0.5). The initial condition was , where
Figure 7.4. Illustration of the coalescing colony model. First, from t = 0, the range of the mother colony (disk) expands by short-distance dispersal with a constant rate c (left). Then (center), the mother colony generates long-distance dispersers to the distance L from its border at rate . The child colonies (circles) expand at rate c until they collide with the mother colony after a period of duration . Finally (right), at the time of coalescence, the range of the blue colony, now including the green one, is immediately reshaped into a circular pattern
Figure 7.5. Pattern of plants that have been detected as infected by Xylella fastidiosa in Corsica between August 2015 and May 2017
Figure 7.6. Map of the Åland Islands and patches of Plantago lanceolata that are healthy (dots) and infected (circles) in 2003 (top) and 2004 (bottom). For a color version of this figure, see www.iste.co.uk/azais/markov.zip
Figure 7.7. Zero-one-inflated posterior distributions of the infection times of six different patches in the year 2004 (top). Locations of patches in the Åland Islands are indicated in the bottom panel. In each top panel, the dots at times zero and one give the posterior probabilities that the infection time is zero and one, respectively. For a color version of this figure, see www.iste.co.uk/azais/markov.zip
Figure 7.8. Paths in the plane of a standard Brownian motion starting at point (0,0) marked by “1” and arriving at an unconditioning point marked by “2” (left) and a standard Brownian bridge starting and arriving at point (0,0) marked by “1” (right)
Figure 7.9. Left: counting Poisson process N(t) (broken line) and its compensator (continuous line) with α = 1 and β = 1.2. Right: the corresponding compensating martingale
Figure 7.10. Realization of 2D martingales , with f₁(t) = 1 + cos(t) and f₂(t) = 0.5 − 2 sin(3t) (left panel), and f₁(t) = cos(0.3t) and f₂(t) = 2 sin(0.3t) (right panel)
Figure 7.11. Realizations of the path obtained with a SDE with jumps (e.g. stochastic differential equations with impulsions). Left: ellipsoidal orbits (β = 2). Right: hyperbolic orbits (β = −0.7)
Figure 7.12. Sample paths of 2D stochastic exponentials Z(t) = ε_M(t) (see equation [7.27]) driven by compensating martingales M_j(t); j = 1, 2, based on a standard Poisson process N(t). Left: and . Right: and

First published 2018 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

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The rights of Romain Azaïs and Florian Bouguet to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Control Number: 2018944661

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

ISBN 978-1-78630-302-8

Preface

The idea for this book stems from the organization of a workshop that took place in Nancy in February 2017. Our motivation was to bring together the French community of statisticians – and a few probability researchers – working directly or indirectly on piecewise-deterministic Markov processes (PDMPs). Thanks to the impetus and advice of Prof. Nikolaos Limnios, we were able to convert this short manifestation into a lasting project, this book.

Since PDMPs form a class of stochastic model with a large scope of applications, many mathematicians have come to work on this subject, sometimes without even realizing it. Although these stochastic models are rather simple, the issue of statistical estimation of the parameters ruling the jump mechanism is far from trivial. The aim of this book is to offer an overview of state-of-the-art methods developed to tackle this issue. Thus, we invited our orators and their co-authors to participate in this project and tried to keep the style of the various authors while providing a homogeneous work with consistent notation and goals.

Statistical Inference for Piecewise-deterministic Markov Processes consists of a general introduction and seven autonomous chapters that reflect the research work of their respective authors, with distinct interests and methods. Nevertheless, they can be investigated according to two reading grids corresponding to the application domains (biology in Chapters 1, 2 and 7, reliability in Chapters 5 and 6, and risk and insurance in Chapters 3 and 4) or to the statistical issues (non-parametric jump rate estimation in Chapters 1 and 2, estimation problems related to level crossing in Chapters 3, 4 and 5, and parametric estimation from partially observed trajectories in Chapters 6 and 7).

The production of this book and of the workshop it originates from would not have been possible without the direct support of the Inria Nancy–Grand Est research center, the Institut Élie Cartan de Lorraine and grants from the French institutions Centre National de la Recherche Scientifique and Agence Nationale de la Recherche.

This adventure started in Nancy, but we write these opening lines half a world apart, each of us far from Lorraine. We want to dedicate this book to all the friends and colleagues we have there. We sincerely thank all the authors, and also the orators of the workshop who did not participate in the writing of this book but nevertheless contributed to a delightful colloquium. Last but not least, warm thanks are due to Marine and Élodie, who constantly encouraged and supported us during this project.

Romain AZAÏS & Florian BOUGUET
May 2018

List of Acronyms

a.s.	almost surely
c.d.f.	cumulative distribution function
càdlàg	right continuous with left limits
CL	Cramér–Lundberg
CLT	central limit theorem
CLVQ	competitive learning vector quantization
DP	diffusion process
EM	expectation maximization
FCP	fatigue crack propagation
i.i.d.	independent and identically distributed
KDEM	kinetic dietary exposure model
MC	Markov chain
MCMC	Markov chain Monte Carlo
ODE	ordinary differential equation
PDE	partial differential equation
PDMP	piecewise-deterministic Markov process
r.v.	random variable
RP	renewal process
SA	Sparre–Andersen
SDE	stochastic differential equation