Dependency and boundary results, some applications in finance and insurance.

Summary This thesis deals with the study of dependencies between risks, using copulas. Taking into account the dependencies that can exist between risks has become crucial for risk managers, and the stakes can be colossal (risk of contagion - and chain bankruptcies - within a portfolio of risky bonds, or correlations between extreme risks in reinsurance - where several a priori independent risks are affected when a catastrophe occurs). A better knowledge of the dependency structures is then fundamental in order to propose an adequate modeling. Therefore, the emphasis in this thesis is on the deformation of copulas as a function of time, or in the tails of the distribution. The first part is devoted to the temporal deformation of copulas in a credit risk context. By introducing conditional copulas, it is thus possible to study the dependence between the lifetimes before default of an issuer, knowing that no default has been observed during a given time period. Fixed point theorems allow to obtain limit behaviors, and to obtain results on first-to-default, for example. The following chapters deal with the use of conditional copulas in the modeling of extreme risks. The theory of extremes in a multivariate framework has been traditionally done by modeling maxima by components. But the study by joint threshold crossing offers a much greater richness. In particular, the study in the upper and lower tail is presented in the case of Archimedean copulas, with emphasis on the characterizations of the cases of asymptotic independence, usually so difficult to apprehend. The last part deals with the nonparametric estimation of copula densities, where kernel estimators are studied, allowing to avoid edge effects traditionally unavoidable when estimating a density with compact support. In particular, techniques are used to correctly estimate the density in the tails of distributions, including with censored data. Finally, a bijgevoegde stelling concludes this thesis on the study of time dependence for climate risks. Long memory models are used to model the risk of storms and to estimate the return period of the August 2003 heat wave. Finally, high-frequency models (similar to those used in finance to model the prices of securities on a transaction-by-transaction basis) are used to model hydrological data, and to propose new estimates for the risk of flooding.