Risk diversification for infinitely divisible distributions

Peng Liu and Tiantian Mao

Papers from arXiv.org

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 L\'{e}vy measures, 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. In particular, we show that the symmetric 1-stable L\'{e}vy process is the only symmetric L\'{e}vy 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 L\'{e}vy measures of the corresponding multidimensional L\'{e}vy processes exhibiting the two diversification phenomena uniformly over time. 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 and the reverse diversification order extend beyond L\'{e}vy processes to several sample-path-dependent processes including running maxima and L\'{e}vy-driven stochastic integrals, with L\'{e}vy-driven Ornstein-Uhlenbeck processes as an important special case. Applications to ruin theory, storage processes and stochastic volatility are also discussed.

Date: 2026-09
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