Risk diversification for infinitely divisible distributions

Abstract

In this paper, we study the diversification properties of convex combinations of iid random variables with infinitely divisible distributions. We characterize, in terms of subadditivity and concavity of the transformed Lévy measures, Lévy processes that exhibit the non-diversification phenomenon or the reverse diversification order with respect to the majorization order uniformly over all time horizons. In particular, we show that the symmetric 1-stable Lévy process is the only symmetric Lévy process exhibiting the non-diversification phenomenon. We further investigate convex combinations of components of multivariate infinitely divisible distributions, allowing for dependent and heterogeneously distributed risks, and characterize the Lévy measures of the corresponding multidimensional Lévy processes exhibiting the two diversification phenomena uniformly over time. Explicit characterizations are obtained for the multidimensional symmetric Lévy processes, multidimensional $α$-stable processes and multidimensional compound Poisson processes. Finally, we show that the non-diversification phenomenon and the reverse diversification order extend beyond Lévy processes to several sample-path-dependent processes including running maxima and Lévy-driven stochastic integrals, with Lévy-driven Ornstein-Uhlenbeck processes as an important special case. Applications to ruin theory, storage processes and stochastic volatility are also discussed.