Journée SMAI-SFdS "Fiabilité et Assurance"
Mercredi 16 novembre 2011 à Toulouse
Cette journée est une initiative conjointe du groupe Modélisation Aléatoire et Statistique (MAS) de la Société de Mathématiques Appliquées et Industrielles (SMAI) et des groupes Banque, Finance et Assurance (BFA) et Incertitudes et Fiabilité (IF) de la Société Française de Statistique (SFdS). L'objectif est de réunir des chercheurs en mathématiques appliquées et des industriels autour de préoccupations communes, à savoir autour de problématiques de risques, de durée de vie, de premiers temps de passage, etc.
This talk is devoted to some stochastic models, inspired by ruin theory, used to describe the phenomenon of exposure to a certain food contaminant. Most recent works in toxicology are mainly centered on static approaches for modelling the quantity of a specific food contaminant ingested on a short period of time and computing the probability that this quantity exceeds a maximum tolerable dose, eventually causing adverse effects on human health In dynamic models, the temporal evolution of the contamination exposure is determined by the accumulation phenomenon due to successive dietary intakes in the one hand and by the pharmacokinetics governing the elimination process in between intakes in the other hand. The exposure dynamic through time is described as a piecewise deterministic Markov process. The problem is then to evaluate the probability that the process crosses some given threshold or to compute some risk functionals associated to the process (length of time over the threshold, severity etc...) . Sample paths of the contamination exposure process are scarcely observable in practice. Intensive computer simulation methods are crucially required for estimating time-dependent or steady-state features of the process. We give some results on the asymptotic properties of simulation estimators based on consumption and contamination data and investigate the problem of constructing accurate bootstrap confidence intervals for certain quantities of importance from the epidemiology viewpoint. We review some applications of these statistical methods to a collection of datasets related to dietary methylmercury (MeHg) contamination.
In the individual risk model, risks often have positively dependent relations. Some notions of positive dependence have been proposed to describe such dependent risks. In this talk, we first discuss the properties of the positive dependence through the stochastic ordering (PDS). Many interesting random vectors have the PDS property such as independent and comonotonic random vectors. Furthermore, the PDS property can also be characterized by copulas. We then show that when the optimization criterion is to minimize the retained risk of an insurer respect to the convex order, the excess-of-loss treaty is the optimal reinsurance form for the insurer among a general class of individualized reinsurance policies with the PDS dependent risks. We also illustrate how to derive the explicit expressions of the optimal excess-of-loss treaty. This talk is based on some recent joint work with Wei Wei.
We show that a simple mixing idea allows to establish a number of explicit formulas for ruin probabilities and related quantities in collective risk models with dependence among claim sizes and among claim inter-occurrence times. Examples include compound Poisson risk models with completely monotone marginal claim size distributions that are dependent according to Archimedean survival copulas as well as renewal risk models with dependent inter-occurrence times.
L'insciption à la journée est gratuite mais obligatoire (places limitées). Elle comprend l'accès aux conférences et les pauses café. Pour s'inscrire, il suffit d'envoyer un email à smai-sfds-tlse[at]univ-pau.fr avant le 30/09/2011, en précisant vos nom, prénom, statut, affiliation et adresse.
Christian Paroissin, Université de Pau et des Pays de l'Adour, Laboratoire de Mathématiques et de leurs applications (LMA).
Contact : smai-sfds-tlse[at]univ-pau.fr