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Investigations on Stochastic Calculus, Statistics of Processes Applied Statistics for Biology and Medical Research Nicolas Savy

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Investigations on Stochastic Calculus, Statistics of Processes Applied Statistics for Biology and Medical Research Nicolas Savy Investigations on Stochastic Calculus, Statistics of Processes Applied Statistics for Biology and Medical Research Nicolas Savy To cite this version: Nicolas Savy. Investigations on Stochastic Calculus, Statistics of Processes Applied Statistics for Biology and Medical Research . Applications [stat.AP]. Université de Toulouse 3 Paul Sabatier, 2014. tel-01645597 HAL Id: tel-01645597 https://tel.archives-ouvertes.fr/tel-01645597 Submitted on 23 Nov 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Document de synth`esepour l'obtention d'une Habilitation `adiriger des recherches Discipline : Math´ematiques Sp´ecialit´e: Probabilit´eset Statistiques Investigations on Stochastic Calculus, Statistics of Processes Applied Statistics for Biology and Medical Research pr´esent´epar Nicolas SAVY au vu des rapports de Pr Michael Kosorok University of North Carolina Pr Marc Lavielle INRIA Saclay Pr David Nualart University of Kansas Soutenue publiquement le 18 juin 2014 devant le jury compos´ede Pr Serge Cohen Universit´ePaul Sabatier Examinateur Pr Fabrice Gamboa Universit´ePaul Sabatier Coordinateur Pr Michael Kosorok University of North Carolina Rapporteur Pr Thierry Lang Universit´ePaul Sabatier Examinateur Pr Marc Lavielle INRIA Saclay Rapporteur Pr Ivan Nourdin Universit´edu Luxembourg Examinateur Universit´ePaul Sabatier - Toulouse III Institut de Math´ematiquesde Toulouse Document de synth`esepour l'obtention d'une Habilitation `adiriger des recherches Discipline : Math´ematiques Sp´ecialit´e: Probabilit´eset Statistiques Investigations on Stochastic Calculus, Statistics of Processes Applied Statistics for Biology and Medical Research pr´esent´epar Nicolas SAVY au vu des rapports de Pr Michael Kosorok University of North Carolina Pr Marc Lavielle INRIA Saclay Pr David Nualart University of Kansas Soutenue publiquement le 18 juin 2014 devant le jury compos´ede Pr Serge Cohen Universit´ePaul Sabatier Examinateur Pr Fabrice Gamboa Universit´ePaul Sabatier Coordinateur Pr Michael Kosorok University of North Carolina Rapporteur Pr Thierry Lang Universit´ePaul Sabatier Examinateur Pr Marc Lavielle INRIA Saclay Rapporteur Pr Ivan Nourdin Universit´edu Luxembourg Examinateur Universit´ePaul Sabatier - Toulouse III Institut de Math´ematiquesde Toulouse Remerciements First of all, I would like to thank David Nualart, Michael Kosorok and Marc Lavielle for accepting to report on my manuscript and for their interest in my work. I am really honoured to see such eminent scientists for my examination board and I warmly thank each of them for their participation. Je suis tr`es heureux que Serge Cohen ait accept´ede prendre place dans ce jury. Il a fait beaucoup lors de mon arriv´ee `aToulouse et s'est toujours montr´ebienveillant et de bon conseil. Je l'en remercie tr`essinc`erement. Sans Thierry Lang, il est certain que la seconde partie de ce m´emoiren'existerait pas. Je suis vraiment combl´e qu'il ai accept´ede participer `ace jury. Ivan Nourdin a accept´emon invitation `aparticiper `ace jury. J'en suis tr`esheureux et lui en suis tr`esreconnaissant. Enfin, un ´enormemerci `aFabrice Gamboa qui a coordonn´emon habilitation avec une grande efficacit´e.Son enthousiasme et son professionalisme m'ont ´et´ed'un grand soutien, Merci. Ensuite, mes remerciements vont `al'ensemble des co-auteurs avec qui j'ai collabor´edurant ces ann´ees. Je voulais tout particuli`erement saluer Vladimir Anisimov, Bernard Bercu, Jean-Fran¸coisDupuy et Josep Vives. Le temps passe, les co-auteurs commencent `a^etrenombreux, je ne les citerai donc pas tous mais je les remercie tr`essinc`erement. Laurent Decreusefond a fait beaucoup pour moi durant ma th`ese mais ´egalement `ala p´eriode charni`ere des premi`eresann´eesen tant que Ma^ıtrede conf´erences,je tiens `ale remercier vivement pour tout ce qu'il m'a appris. Merci ´egalement `atous les membres de l'IMT. Ceux avec qui j'ai eu le plaisir de travailler, Fabien Panloup, Ald´ericJoulin, Monique Pontier, C´ecileChouquet... Ceux avec qui il est toujours plaisant de discuter, Patrick Cattiaux, Laurent Miclo, Jean-Michel Loub`es,Thierry Klein, Jean-Marc Aza¨ıs,Jean- Yves Dauxois, Didier Concordet... Celui avec qui je partage - avec un grand plaisir - mon bureau depuis mon arriv´ee`aToulouse, S´ebastienGadat. Enfin je remercie tout sp´ecialement Laure Coutin pour toute l'aide qu'elle m'a apport´ee,pour sa gentillesse et sa bienveillance. Je n'oublie pas le personnel admin- istratif et technique de l'IMT qui fait un travail remarquable. Merci Fran¸coise,Marie-Laure, Karima, Tamara, Marie-Line, Agn`eset les informaticiens. Travailler `al'interaction Math´ematiques/ Sciences du vivant ne peut se faire sans une relation ´etroite entre des ´equipes de chacune des disciplines. Ce n'est pas forc´ement facile, les codes, le langage ne sont pas toujours les m^emes. Pour ma part, j'estime avoir beaucoup de chance de travailler avec les ´equipes que je tiens `aremercier maintenant. • Au premier chef, l'unit´eINSERM 1027 et notamment l'´equipe 1 de Sandrine Andrieu et l'´equipe 5 de Thierry Lang. C'est un grand plaisir de travailler avec ces deux ´equipes. J'en profite pour saluer la remarquable ouverture d'esprit des membres de ces ´equipes et pour remercier comme il se doit Sandrine Andrieu. • Le cercle des collaborations s'est aggrandi jusqu’`al'´equipe 6 de l'unit´eINSERM 1027. Autour de la th`esede Caroline Delarue, avec Christine Damase, Isabelle Lacroix et C´ecileChouquet nous navigons entre statistiques et pharmacologie... c'est un vrai plaisir. • Ensuite, un grand merci `aAntoine Blancher, chef du service d'Immunologie du CHU de Toulouse, qui me fait d´ecouvrirles m´eandresde la g´en´etique. • Merci ´egalement a Felipe Guerrero du laboratoire d'H´ematologiedu CHU de Toulouse. • Enfin, le canc´eropleGSO par le biais d'Olivier Claverie a souvent ´et´el`apour financer des ´ev´enements que j'ai organis´eet pour encourager des activit´esde recherches, je l'en remercie. 1 Je tiens `afaire un paragraphe sp´ecialpour les ´etudiants que j'ai eu le plaisir d'encadrer. Je leur dois beaucoup. Merci `aCaroline, Merci `aNathan et Billy actuellement en th`eseet surtout un ´enormemerci `aGuillaume, Beno^ıtet Val´eriequi ont soutenu leur th`ese.Je suis fier d'eux et je leur souhaite beaucoup de bonheur. Je remercie ´egalement mes coll`eguesdu d´epartement GEA-R de l'IUT de Toulouse. Merci notamment `aFabien, toujours disponible pour arranger nos emplois du temps, Marc mon coll`egue de bureau, Nad`ege et Marie-Th´er`eseles chefs de d´epartement, Yann, Andr´e,Marc, Nathalie, Ann et tous les enseignants du d´epartement. Bien s^uremerci ´egalement `aSylvie, Marjorie, Jocelyne et Franck de nous faciliter la vie au quotidien. Je n'oublie pas mes parents, mes fr`ereset plus g´en´eralement toute ma famille et ma belle-famille. Une pens´eechaleureuse `atous. Mes plus profonds remerciements vont `aSt´ephanie. Sans son enthousiasme communicatif pour les sciences du vivant et sa facult´e`asoulever des questions passionnantes, la seconde partie de ce m´emoire n'aurait jamais vu le jour. Merci ´egalement pour le soutien au quotidien. Merci pour les encouragements. Merci pour les conseils... Bref, Merci pour tout et bien plus encore. Je finirai le difficile exercice des remerciements par le plus simple : Gros bisous `aL´eonelet Achille. 2 Contents Introduction 5 Bibliography . .8 A- Investigations on Stochastic Calculus and Statistics of Processes 11 1 Malliavin Calculus and Anticipative Integrals 13 I Filtered Poisson Process . 13 II Skohorod Integral with respect to filtered Poisson process . 16 III Anticipative integral for filtered L´evyprocess . 22 Bibliography . 28 2 A limit theorem for filtered Poisson processes 31 I Hilbertian martingale and Radonification . 31 II Radonification of martingales associated to filtered processes . 32 III Convergence of Hilbertian martingales . 33 IV Back to the initial problem . 33 Bibliography . 34 3 Transportation Inequality and Malliavin Calculus 35 I Ingredients . 35 II Upper bounds on Rubinstein distances . 38 III Applications . 38 Bibliography . 41 4 Properties of Estimators for some diffusion processes. 43 I Estimator of instantaneous volatility . 44 II Large deviation Principle for drift parameter . 48 III Ornstein-Uhlenbeck driven by Ornstein-Uhlenbeck processes . 61 Bibliography . 63 B- Statistic applied to Biology and to Medical Research 65 5 Models for patients' recruitment in clinical trials 67 I General considerations. 67 II The Bayesian-Poisson models and their performance . 71 III Integration of screening-failures . 73 IV An additive model for clinical trials' cost modelling [13] . 78 Bibliography . 80 6 Survival data analysis for prevention in RCT 81 I Weighted logrank tests . 82 II Fleming-Harrington's test . 85 III An "omnibus" test: maximum weighted Logrank statistic . 89 IV Application: the GuidAge study . 94 Bibliography . 95 3 CONTENTS 7 Elements for mediation and evolution in Epidemiology 97 I Approach by Markov Models . 98 II Approach by Causal Structural Equations . 101 Bibliography . 106 8 Various results in interaction with Biology 109 I Predictive microbiology . 109 II Population Genetics . 112 Bibliography . 113 C- Conclusions and Perspectives 115 9 Conclusions and Perspectives 117 I On stochastic calculus . 117 II On survival data analysis . 118 III On patients' recruitment modelling . 118 IV On Clinical Trials Simulation . 119 V On mediation analysis . 120 VI On Prediction in Social Health . 121 Bibliography . 121 Curriculum Vitae 122 4 Introduction My research activity divides into two really distinct areas: probability and mathematical statistics of stochastic processes and applied statistics for Medicine and Biology.
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