Convex extrema for nonincreasing discrete distributions: effects of convexity constraints

In risk management, the distribution of underlying random variables is not always known. Sometimes, only the mean value and some shape information (decreasingness, convexity after a certain point,...) of the discrete density are available. The present paper aims at providing convex extrema in some cases that arise in practice in insurance and in other fields. This enables us to obtain for example bounds on variance and on Solvency II related quantities in insurance applications. In this paper, we first consider the class of discrete distributions whose probability mass functions are nonincreasing on a support ${\cal D}_n\equiv \{0,1,\ldots,n\}$. Convex extrema in that class of distributions are well-known. Our purpose is to point out how additional shape constraints of convexity type modify these extrema. Three cases are considered: the p.m.f. is globally convex on $\N$, it is convex only from a given positive point $m$, or it is convex only up to some positive point $m$. The corresponding convex extrema are derived by using simple crossing properties between two distributions. The influence of the choice of $n$ and $m$ is discussed numerically, and several illustrations to ruin problems are presented. These results provide a complement to two recent works by Lefévre and Loisel (2010), (2012).