On some negative dependence structures and their applications

Abstract

Recently, the study of negative dependence structures has aroused considerable interest amongst researchers in actuarial science and quantitative risk management. This thesis centres on two extreme negative dependence structures in different dimensions - counter-monotonicity and mutual exclusivity, and develops their novel characterizations and applications to risk management.

Bivariate random vectors are treated in the first part of the thesis, where the characterization of comonotonicity by the optimality of aggregate sums in convex order is extended to its bivariate antithesis, namely, counter-monotonicity. It is shown that two random variables are counter-monotonic if and only if their aggregate sum is minimal with respect to convex order. This defining property of counter-monotonicity is then exploited to identify a necessary and sufficient condition for merging counter-monotonic positions to be risk-reducing.

In the second part, the notion of mutual exclusivity is introduced as a multi-dimensional generalization of counter-monotonicity. Various characterizations of mutually exclusive random vectors are presented, including their pairwise counter-monotonic behaviour, minimal convex sum property, and the characteristic function of their aggregate sums. These properties highlight the role of mutual exclusivity as the strongest negative dependence structure in a multi-dimensional setting. As an application, the practical problem of deriving general lower bounds on three common convex functionals of aggregate sums with arbitrary marginal distributions is considered. The sharpness of these lower bounds is characterized via the mutual exclusivity of the underlying random variables. Compared to existing bounds in the literature, the new lower bounds proposed enjoy the advantages of generality and simplicity.