Risk process: dependency modeling and risk assessment under convexity constraints.

Summary This thesis focuses on two different issues that have in common the contribution to modeling and risk management in actuarial science. In the first research theme, we are interested in the modeling of dependence in insurance and in particular, we propose an extension of the common factor models used in insurance. In the second research theme, we consider decreasing discrete distributions and study the effect of the addition of the convexity constraint on the convex extrema. Applications in connection with the theory of ruin motivate our interest in this subject. In the first part of the thesis, we consider a discrete time risk model in which the random variables are dependent but conditionally independent with respect to a common factor. In this dependence framework, we introduce a new concept for modeling the time dependence between risks in an insurance portfolio. Indeed, our modeling includes unbounded memory processes. More precisely, the conditioning is done with respect to a random vector of variable length over time. Under conditions of mixing the factor and a conditional mixing structure, we have obtained mixing properties for unconditional processes. With these results we can obtain interesting asymptotic properties. We note that in our asymptotic study it is rather the time that tends to infinity than the number of risks. We give asymptotic results for the aggregate process, which allows us to approximate the risk of an insurance company when time tends to infinity. The second part of the thesis deals with the effect of the convexity constraint on the convex extrema in the class of discrete distributions whose probability mass functions (p.m.f.) are decreasing on a finite support. The convex extrema in this class of distributions are well known. Our goal is to highlight how additional shape constraints of the convex type modify these extrema. Two cases are considered: the p.m.f. is globally convex on N and the p.m.f. is convex only from a given positive point. The corresponding convex extrema are computed using simple crossing properties between two distributions. Several illustrations in ruin theory are presented.