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Speed of Convergence of Diffusion Approximations Eustache Besançon Speed of convergence of diffusion approximations Eustache Besançon To cite this version: Eustache Besançon. Speed of convergence of diffusion approximations. Probability [math.PR]. Institut Polytechnique de Paris, 2020. English. NNT : 2020IPPAT038. tel-03128781 HAL Id: tel-03128781 https://tel.archives-ouvertes.fr/tel-03128781 Submitted on 2 Feb 2021 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. Speed of Convergence of Diffusion Approximations Thèse de doctorat de l’Institut Polytechnique de Paris T038 préparée à Télécom Paris École doctorale n°574 École doctorale de mathématiques Hadamard (EDMH) : 2020IPPA : Spécialité de doctorat: Mathématiques Appliquées NNT Thèse présentée et soutenue à Paris, le 8 décembre 2020, par Eustache Besançon Composition du Jury: Laure Coutin Professeur, Université Paul Sabatier (– IM Toulouse) Président Kavita Ramanan Professor, Brown University (– Applied Mathematics) Rapporteur Ivan Nourdin Professeur, Université du Luxembourg (– Dpt Mathématiques) Rapporteur Philippe Robert Directeur de Recherche, INRIA Examinateur François Roueff Professeur, Télécom Paris (– LTCI) Examinateur Laurent Decreusefond Professeur, Télécom Paris (– LTCI) Directeur de thèse Pascal Moyal Professeur, Université de Lorraine (–Institut Elie Cartan) Directeur de thèse Speed of convergence of diffusion approximations Eustache Besan¸con December 8th, 2020 2 Contents 1 Introduction 7 1 Diffusion Approximations . .7 2 Fields of Application . .8 2.1 Queueing theory . .8 2.2 Epidemiology . 11 2.3 Genetics . 14 3 Our contribution: a general approach to the rate of convergence inlaw ................................. 16 2 Roadmap 19 1 The goal . 19 2 Overview of the Stein Method . 20 3 Roadmap . 23 3 Choice of a functional space 25 1 Fractional Sobolev spaces . 25 2 Wiener structure on a fractional Sobolev space . 27 4 Stein Method in Wiener fractional Sobolev Spaces 37 1 Malliavin calculus for Poisson processes . 37 2 Stein Operator and linear interpolation . 41 5 Application to a class of Markov Processes 49 1 A class of Markov processes . 49 2 The scaling of the processes . 50 3 Interpolation of these processes . 53 3.1 Distance between the process and its interpolation . 53 3.2 Distance between two affine interpolations . 54 4 A central limit theorem . 56 6 Application to Queueing Models 61 1 The M=M=1 queue . 61 1.1 Short introduction to the M=M=1 model . 61 1.2 Straight application of our model . 63 2 The M=M=1 model . 64 2.1 Short introduction to the M=M=1 model . 64 2.2 Higher rate of convergence for a smaller class of functions 66 3 7 Other Applications 69 1 The ON/OFF model . 69 2 The SIR model . 71 3 The Moran model with selection . 73 8 Appendix 75 1 Besov-Liouville Spaces . 75 2 Moment bound for Poisson variables . 79 Aknowledgement All my thanks go to professors Laurent Decreusefond and Pascal Moyal, who were kind enough to accept to de supervise my doctoral thesis, and had the intelligence to invent a fascinating subjectas well as the patience to guide me in my research work. Without them this work would not have been possible. I thank Kavita Ramanan and Ivan Nourdin who have accepted to review my work and act as referees. I thank also professors Laure Coutin and Philippe Robert, who have accepted to be part of the jury and last but not least, pro- fessor Fran¸coisRoueff who greeted me in the laboratory, and guided me in the administrative arcanes of the school. I finally thank for their patience my wife Olga and my daughters Juliette and Enora whom I did not attend as much as expected while I was writing this thesis. R´esum´e Dans de nombreux champs d'applications (files d'attentes, ´epid´emiologie,g´en´etique, finance), les processus de Markov sont un outil privil´egi´ede mod´elisationde processus al´eatoires. Malheureusement, il est souvent n´ecessaired'avoir re- cours `ades espaces d'´etats tr`esgrands voire infinis, rendant l'analyse exacte des diff´erentes caract´eristiques(stabilit´e,loi stationnaire, temps d'atteinte de certains domaines, etc.) du processus d´elicateou impossible. Depuis longtemps, gr^acenotamment `ala th´eoriedes martingales, on proc`ede`ades approximations par des diffusions browniennes. Celles-ci permettent souvent une analyse ap- proch´eedu mod`eled'origine. Le principal d´efautde cette approche est que l'on ne conna^ıtpas l'erreur com- mise dans cette approximation. Il s'agit donc ici de d´evelopper une th´eorie du calcul d'erreur dans les approximations diffusion. Depuis quelques temps, le d´eveloppement de la m´ethode de Stein-Malliavin a permis de pr´eciserles vitesses de convergence dans les th´eor`emesclassiques comme le th´eor`emede Donsker (convergence fonctionnelle d'une marche al´eatoirevers un mouvement brown- ien) ou la g´en´eralisationtrajectorielle de l'approximation binomiale-Poisson. Il s'agit dans ce travail de poursuivre le d´eveloppement de cette th´eoriepour des processus de Markov comme ceux que l'on rencontre en th´eoriedes files d'attente ou en ´epid´emiologieet dans bien d'autres domaines appliqu´es. Or tous les travaux pr´ec´edents utilisant la m´ethode de Stein en dimension infinie ont ´etudi´ela convergence de processus vers des mouvements Browniens stan- dards ou vers des processus de Poisson alors que les limites fluides des approxi- mations diffusions peuvent prendre des formes vari´ees.Il s'agit donc d'inventer 4 une mani`ered'appliquer la m´ethode de Stein `aun ensemble plus large de situ- ations. On se situe d'embl´eedans un espace de dimension infinie : les espaces de Sobolev fractionnaires, qui sont suffisamment larges pour comprendre les fonctions con- tinues H¨olderiennesque sont les mouvements Browniens et les fonctions con- stantes par morceaux que sont les processus de Poisson, et qui pr´esentent des car- act´eristiquesfort commodes facilitant les calculs (espaces de Banach s´eparables , existence d'une injection canonique dans un espace de Hilbert permettant d'utiliser un produit scalaire notamment). Partant de la repr´esentation des pro- cessus de Markov comme mesures de Poisson, on ´etendla m´ethode d´evelopp´ee par Laurent Decreusefond et Laure Coutin pour estimer la vitesse de conver- gence dans les approximations diffusion. On munit ces espaces de Sobolev d'une mesure de Wiener abstraite et l'on construit un processus d'Ornstein Uhlenbeck en dimension infinie, ayant pour mesure invariante la mesure de Wiener. On est alors en mesure d'exprimer l'´equationde Stein caract´erisant la convergence vers le mouvement Brownien en utilisant les travaux de Shih. Cette ´equationreste valide pour toute fonctionnelle Lipschitzienne. Afin d'exploiter cette ´equation, il faut utiliser le calcul de Malliavin sur les processus de Poisson, et notamment la formule d'int´egrationpar parties. On ´etendla m´ethode de Stein-Malliavin `ades vecteurs de processus d´ependants plut^otqu’`aun seul processus. Si la suite de processus converge vers une limite pouvant ^etreexprim´eecomme limite d'une ´equationdiff´erentielle `aparam`etres d´eterministes,la limite est un processus Gaussien chang´ede temps. La m´ethode de Stein Malliavin ´etant d´evelopp´eesurtout pour montrer la convergence vers le mouvement Brownien standard, on l'adapte `ala convergence vers un pro- cessus chang´ede temps `atravers des m´ethodes d'approximations lin´eaires,le processus chang´ede temps ´etant approximativement un mouvement Brownien standard sur un petit espace de temps. On utilise deux ´echelles de temps afin d'approcher au mieux le processus discret de d´epartet le mouvement continu `ala limite. Un calcul d'optimisation permet enfin d'obtenir l'´echelle de temps la plus appropri´eepour ´evaluer la vitesse de convergence au plus juste. Celle s'av`ere^etreen d´efinitive de n−1=6 ln n. Cette m´ethodologie est alors appliqu´ee`aquatre exemples d'approximations diffusion : en th´eoriedes file d'attentes, les files M=M=1 et M=M=1. En ´epid´emiologiele mod`eleSIR de cat´egorisationdes populations entre suscepti- bles, infect´eset remis. En g´en´etiquele mod`eleOn/Off. Dans tous ces cas, la vitesse de convergence peut ^etreborn´eepar la quantit´e n−1=6 ln n. `aun facteur multiplicatif constant pr`es.Cette m´ethode n'est toutefois pas universelle, et le mod`elede Moran, utilis´een g´en´etiquedes populations, ne se pr^etepas `ason utilisation, car il ne converge pas vers une ´equationdiff´erentielle `aparam`etres d´eterministes.D'autres m´ethodes restent `ad´ecouvrirpour d´eterminersa vitesse de convergence. 5 6 Chapter 1 Introduction 1 Diffusion Approximations In the present manuscript we shall be interested in particular classes of stochastic models and their asymptotic behaviour. A stochastic process is just a model for a random quantity that evolves through time. Such a mathematical object is a collection of random variables fXt : t 2 Tg indexed by a set T which we can interpret as a set of times. It will often be denoted simply by X in the sequel. In order to denote the value of the process X at time t, we shall use either X(t) or Xt, depending on whether adding an index or parenthesis to the notation makes it easier to read, and these two notations should be considered as equivalent. For us T will always be a compact interval [0;T ] in R+ (continuous time setting).
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