Polynomial approximations of probability densities and applications in insurance.

Summary The purpose of this thesis is to study numerical methods for approximating the probability density associated with random variables that have compound distributions. These random variables are commonly used in actuarial science to model the risk borne by a portfolio of contracts. In catastrophe theory, the probability of ultimate catastrophe in the compound Poisson model is equal to the survival function of a compound geometric distribution. The proposed numerical method consists of an orthogonal projection of the density onto a basis of orthogonal polynomials. These polynomials are orthogonal with respect to a reference probability measure belonging to the Quadratic Natural Exponential Families. The polynomial approximation method is compared to other density approximation methods based on moments and the Laplace transform of the distribution. The extension of the method in dimension higher than $1$ is presented, as well as the obtention of a density estimator from the approximation formula. This thesis also includes the description of an aggregation method adapted to portfolios of life insurance contracts of individual savings type. The aggregation procedure leads to the construction of model points to allow the evaluation of the best estimate reserves in a reasonable time and in accordance with the European Solvency II directive.