Processes and risk indicators in non-life insurance and food safety.

Summary Risk analysis has become a major issue in our society. Whatever the field of application in which a risky situation may occur, mathematics, and more particularly statistics and probabilities, are essential tools. The main purpose of this thesis is to develop relevant risk indicators and to study the extremal properties of processes involved in two fields of application: food risk and insurance. Risk theory is situated between extreme value analysis and the theory of random variables with regular variations or heavy tails. In the first chapter, we define the key elements of the risk theory as well as the notion of regular variation and we introduce different models related to food risk which will be studied in chapters 2 and 3. Chapter 2 presents the work done with Olivier Wintenberger. For classes of stochastic processes, under assumptions of regular variation, we develop a method that allows us to obtain asymptotic equivalents in a finite horizon of risk indicators in insurance and food risk such as the probability of ruin, the "time spent above a threshold" or the "severity of the ruin". Chapter 3 focuses on food risk models. Specifically, we study the extremal properties of different generalizations of a contaminant exposure process named KDEM for Kinetic Dietary Exposure Model proposed by Patrice Bertail and his co-authors in 2008. Under assumptions of regular variations, we propose asymptotic equivalents of the tail behavior and the extremal index of the exposure process. Finally, Chapter 4 reviews different statistical techniques particularly suited to study the extremal behavior of some Markov processes. Thanks to regenerative properties, it is possible to split the path of observations into independent and identically distributed blocks and thus study only the process on a block. These techniques apply even if the Markov chain is not atomic. We focus here on the estimation of the tail index and the extremal index. We illustrate the performance of these techniques by applying them on two models - in insurance and in finance - for which we know the theoretical results.