Risk diversification for infinitely divisible distributions

In this paper, we study the diversification properties of convex combinations of iid infinitely divisible random variables. For L\'{e}vy processes with bounded variation sample paths, we characterize, in terms of subadditivity and concavity of the transformed L\'{e}vy tails, L\'{e}vy processes that exhibit the non-diversification phenomenon or the reverse diversification order with respect to the majorization order uniformly over all time horizons. For general symmetric L\'{e}vy processes without a Gaussian component, we show that the symmetric 1-stable L\'{e}vy process is the only nontrivial process exhibiting either phenomenon. We further investigate convex combinations of components of multivariate infinitely divisible distributions, allowing for dependent and heterogeneous components, and characterize the L\'{e}vy measures of multidimensional L\'{e}vy processes exhibiting the two adverse diversification phenomena uniformly over all time horizons. Explicit characterizations are obtained for the multidimensional symmetric L\'{e}vy processes, multidimensional $\alpha$-stable processes and multidimensional compound Poisson processes. Finally, we show that the non-diversification phenomenon extends beyond L\'{e}vy processes to running maxima and integrals of increasing convex functionals of L\'{e}vy processes, while both adverse diversification phenomena are preserved for L\'{e}vy-driven stochastic integrals with nonnegative deterministic kernels. Applications to ruin theory, storage processes and stochastic volatility are also discussed.