Multivariate risk measures and applications in actuarial science.

Summary The entry into force on January 1, 2016 of the European regulatory reform for the insurance sector, Solvency 2, is a historic event that will radically change risk management practices. It is based on an important consideration of the risk profile and vision, via the possibility of using internal models to calculate solvency capital and the ORSA (Own Risk and Solvency Assessment) approach for internal risk management. Mathematical modeling is thus an indispensable tool for a successful regulatory exercise. Risk theory must be able to accompany this development by proposing answers to practical problems, notably related to the modeling of dependencies and the choice of risk measures. In this context, this thesis presents a contribution to the improvement of actuarial risk management. In four chapters we present multivariate risk measures and their application to solvency capital allocation. The first part of this thesis is devoted to the introduction and study of a new family of elicitable multivariate risk measures that we will call multivariate expectations. The first chapter presents these measures and explains the different approaches used to construct them. The multivariate expectations verify a set of consistency properties that we also discuss in this chapter before proposing a tool for the stochastic approximation of these risk measures. The performances of this method being insufficient in the vicinity of the asymptotic levels of the thresholds of the expectiles, the theoretical analysis of the asymptotic behavior is necessary, and will be the subject of the second chapter of this part. The asymptotic analysis is carried out in a multivariate regular variation environment, it allows to obtain results in the case of equivalent marginal tails. We also present in the second chapter the asymptotic behavior of the multivariate expectiles under the previous assumptions in the presence of perfect dependence, or asymptotic independence, and we propose with the help of extreme value statistics estimators of the asymptotic expectile in these cases. The second part of the thesis focuses on the problem of solvency capital allocation in insurance. It is composed of two chapters in the form of published articles. The first one presents an axiomatization of the consistency of a capital allocation method in the most general framework possible, then studies the consistency properties of an allocation approach based on the minimization of multivariate risk indicators. The second paper is a probabilistic analysis of the behavior of the latter allocation approach as a function of the nature of the marginal risk distributions and the dependence structure. The asymptotic behavior of the allocation is also studied and the impact of the dependence is illustrated by different marginal models and different copulas. The presence of dependence between the different risks borne by insurance companies makes the multivariate approach a more appropriate answer to the different problems of risk management. This thesis is based on a multidimensional view of risk and proposes measures of a multivariate nature that can be applied to different actuarial problems of this nature.