Dependency models in risk theory.

Summary Initially, risk theory assumed independence between the different random variables and other parameters involved in actuarial modeling. Nowadays, this assumption of independence is often relaxed in order to take into account possible interactions between the different elements of the models. In this thesis, we propose to introduce dependence models for different aspects of risk theory. First, we suggest the use of copulas as a dependency structure. We first address a Tail-Value-at-Risk capital allocation problem for which we assume a copula-introduced link between different risks. We obtain explicit formulas for the capital to be allocated to the whole portfolio as well as the contribution of each risk when we use the Farlie-Gumbel-Morgenstern copula. For the other copulas, we provide an approximation method. In the second chapter, we consider the random process of the sum of the present values of the claims for which the random variables of the amount of a claim and the time elapsed since the previous claim are linked by a Farlie-Gumbel-Morgenstern copula. We show how to obtain explicit forms for the first two moments and then the mth moment of this process. The third chapter assumes another type of dependence caused by an external environment. In the context of the study of the probability of ruin of a reinsurance company, we use a Markovian environment to model the underwriting cycles. We first assume deterministic cycle phase change times and then consider them influenced in turn by the amounts of claims. Using the erlangization method, we obtain an approximation of the probability of ruin in finite time.