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Generalized Ornstein-Uhlenbeck Processes in Ruin Theory Jerôme Spielmann

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Generalized Ornstein-Uhlenbeck Processes in Ruin Theory Jerôme Spielmann Generalized Ornstein-Uhlenbeck Processes in Ruin Theory Jerôme Spielmann To cite this version: Jerôme Spielmann. Generalized Ornstein-Uhlenbeck Processes in Ruin Theory. Probability [math.PR]. Université d’Angers, 2019. English. NNT : 2019ANGE0066. tel-03148887 HAL Id: tel-03148887 https://tel.archives-ouvertes.fr/tel-03148887 Submitted on 22 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. THESE DE DOCTORAT DE L'UNIVERSITE D'ANGERS COMUE UNIVERSITE BRETAGNE LOIRE ECOLE DOCTORALE N° 601 Mathématiques et Sciences et Technologies de l'Information et de la Communication Spécialité : Mathématiques et leurs interactions Jérôme SPIELMANN Generalized Ornstein-Uhlenbeck Processes in Ruin Theory Thèse présentée et soutenue à Angers, le 13 décembre 2019 Unité de recherche : LAREMA UMR CNRS 6093 Thèse N° : 181271 Rapporteurs avant soutenance : Youri Kabanov Professeur, Université de Franche-Comté Jostein Paulsen Professor, Københavns Universitet Composition du Jury : Loïc Chaumont Professeur, Université d’Angers Alexander Lindner Professor, Universität Ulm Directrice de thèse Lioudmila Vostrikova Professeur, Université d’Angers These` en vue de l'obtention du grade de Docteur en Mathematiques´ appliquees´ pr´esent´ee `al'Universit´ed'Angers par Jer´ ome^ Spielmann Generalized Ornstein-Uhlenbeck Processes in Ruin Theory soutenue le 13 d´ecembre 2019 devant le jury compos´ede Youri Kabanov Rapporteur Professeur, Universit´ede Franche-Comt´e Jostein Paulsen Rapporteur Professeur, Københavns Universitet Lo¨ıc Chaumont Membre du jury Professeur, Universit´ed'Angers Alexander Lindner Membre externe du jury Professeur, Universit¨atUlm Lioudmila Vostrikova Directrice de th`ese Professeur, Universit´ed'Angers Abstract This thesis is concerned with the study of Generalized Ornstein-Uhlenbeck (GOU) processes and their application in ruin theory. The GOU processes, which are the solutions of certain linear stochastic differential equations, have been introduced in ruin theory by Paulsen in 1993 as models for the surplus capital of insurance companies facing both insurance and market risks. In general, these processes were chosen as suitable models on an a priori basis. The first and main contribution of this thesis is to show that GOU processes appear naturally as weak limits of random coefficient autoregres- sive processes which are used extensively in various domains of applied prob- ability. Using this result, the convergence in distribution of the ruin times, the convergence of the ultimate ruin probability and the moments are also shown. The rest of the thesis deals with the study of certain properties of GOU pro- cesses. In particular, the ruin problem for the GOU process is studied and new bounds on the ruin probabilities are obtained. These results also general- ize some known upper bounds, asymptotic results and conditions for certain ruin to the case when the market risk is modelled by a semimartingale. The final section of the thesis moves away from classical ruin theory and lays some first directions for the study of the law of GOU processes at fixed times. In particular, a partial integro-differential equation for the density, large and small-time asymptotics are obtained for these laws. This shift away from the ruin probability is explained by the fact that most risk measures used in practice such as Value-at-Risk are based on these laws instead. Resum´e Cette th`esecontribue `al'´etudedes processus d'Ornstein-Uhlenbeck g´en´eralis´es (GOU) et de leurs applications en th´eoriede la ruine. Les processus GOU, qui sont les solutions de certaines ´equationsdiff´erentielles stochastiques lin´eaires, ont ´et´eintroduits en th´eoriede la ruine par Paulsen en 1993 en tant que mod`eles pour le capital d'une assurance soumise au risque de march´e. En g´en´eral,ces processus ont ´et´echoisis comme mod`elesde mani`ere a priori. La premi`ereet principale contribution de cette th`eseest de montrer que les processus GOU apparaissent de mani`erenaturelle comme limites faibles de processus autoregressifs `acoefficients al´eatoires, processus qui sont tr`es utilis´esen probabilit´eappliqu´ee. A` partir de ce r´esultat,la convergence en distribution des temps de ruine, la convergence des probabilit´esde ruine ainsi que la convergence des moments sont aussi d´emontr´ees. Le reste de la th`esetraite de certaines propri´et´esdes processus GOU. En particulier, le probl`emede la ruine est trait´eet de nouvelles bornes sur les probabilit´esde ruine sont obtenues. Ces r´esultatsg´en´eralisent aussi des r´esultatsconnus au cas o`ule risque de march´eest mod´elis´epar une semi- martingale. La derni`ere partie de la th`eses'´eloignede la th´eoriede la ruine pour passer `al'´etudede la loi du processus `atemps fixe. En particulier, une ´equation int´egrodiff´erentielle partielle pour la densit´eest obtenue, ainsi que des ap- proximations pour la loi en temps courts et longs. Cet ´eloignement s'explique par le fait que la plupart des mesures de risque utilis´ees dans la pratique sont bas´eessur ces lois et non sur la probabilit´ede ruine. Remerciements Tout d'abord, j'aimerais remercier Lioudmila qui a supervis´ema th`eseavec bienveillance, patience et enthousiasme. Je suis honor´ed'avoir pu ´etudier avec elle et d'avoir pu profiter de ses connaissances profondes du calcul stochastique et de la th´eoriedes martingales, de ses intuitions et de sa rigueur math´ematique.Je consid´ereraisque ces trois ann´eesauront ´et´eun succ`essi j'ai pu capter une petite partie des ces qualit´es. J'aimerais ensuite remercier Youri Kabanov et Jostein Paulsen pour avoir accept´ede rapporter ma th`eseainsi qu'Alexander Lindner et Lo¨ıcChaumont pour avoir accept´ede participer `amon jury. Je n'aurais jamais pu faire cette th`esesans Michael Schmutz qui m'a mis con- tact avec Lioudmila et qui a ´eveill´emon int´er^etpour la finance math´ematique durant ses cours du vendredi matin (8h...) `al'EPFL. Je le remercie pro- fond´ement. Je remercie aussi Yuchao et Runsheng avec qui j'ai eu le plaisir de collaborer et de discuter de nombreuses heures. J'aimerais ensuite remercier mes coll`eguesdu LAREMA. En particulier, je remercie Fabien, Fr´ed´eric,Lo¨ıcet Rodolphe qui m'ont beaucoup appris `atravers leurs cours doctoraux et les discussions autour de la machine `a caf´eet/ou d'un tableau et/ou d'un filet de badminton. Je remercie aussi Jacquelin et Fran¸cois qui m'ont aid´eles nombreuses fois o`uj'ai essay´ede faire "sudo" sur ma machine, ainsi que Mohammed et Etienne qui ´etaient toujours disponibles pour r´epondre `ames questions. Ces trois ann´eesau- raient ´et´ebien moins agr´eables,si je n'avais pas rencontr´eSinan qui a tou- jours partag´emes coups de gueule (parfois autour d'une pinte de stout). Je remercie aussi Alexandra et Rahma pour tout leur travail et leur aide pour mes divers probl`emesadministratifs. Finalement, je remercie aussi Denis qui ´egayait notre bureau chaque semaine. J'aimerais ensuite remercier les doctorants et doctorantes qui sont pass´esau LAREMA avant moi. D'abord, je remercie Johan qui m'a beaucoup soutenu quand j'´etaisencore un petit nouveau et qui m'a fait d´ecouvrirle "Rabelais" nantais. Je remercie aussi Ann pour tous les bons moments pass´es`aparler de musique, de politique et de math´ematiques. Je n'oublie bien s^urpas Cl´ement et Simon que je remercie pour de nombreuses soir´eesde jeux et de d´etente. Je remercie aussi Alex qui est arriv´ela m^emeann´eeque moi et avec qui j'ai pu partager mes frustration et mes r´eussites. La th`eseest une ´epreuve difficile et je suis content d'avoir pu discuter avec quelqu'un qui passait par les m^emes´etapes que moi. Puis, viennent les doctorants et doctorantes qui sont arriv´esapr`esmoi. Tout d'abord, je remercie Axel pour les parties de Dominion, les "goudales" partag´eeset les discussions passionnantes au labo et en dehors. Je remercie aussi Ouriel qui m'a fait d´ecouvrirBrassens, Lino et le bridge, Marine avec qui on a r´eussi`amaintenir la r´eputationdes probabilistes face aux attaques des g´eom`etreset autres alg´ebristeset Th´eoqui m'a propos´eun tour en moto que j'ai l^achement refus´e.Je remercie aussi Sofia et Eunice qui contribuent `a la bonne ambiance qui r`egnedans notre open-space du I104. Je leur souhaite du courage pour la th`eseainsi qu’`aAntoine, David, Fran¸cois,Jacob, Marius et Maxime. Je tiens aussi `aremercier du fond du cœur mes amis d'Angers (surtout Nico M.), de Paris, de Gen`eve, de Lausanne, de Zurich, de Berne et de Neuch^atel qui comptent beaucoup pour moi. En particulier, Arnaud, Gabriel, Ga¨el, Nico, No´emie,Mika, Thibaut, Yohan, mes cousins et cousines que j'ai trop peu vus durant ces trois derni`eresann´ees. Je ne serais pas la sans l'amour de mes parents Angela et Daniel et de ma sœur C´eline. Je les remercie simplement pour tout. Je remercie aussi le reste de ma famille qui m'ont toujours soutenu. Finalement, je remercie Laura pour son amour qui illumine chaque instant de ma vie. Contents Introduction 13 1 Mathematical Definitions and Basic Facts 19 1.1 Preliminaries and Notations . 19 1.2 Semimartingales and Stochastic Calculus .
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