Risk diversification for infinitely divisible distributions
Abstract
In this paper, we study the diversification properties of convex combinations of iid infinitely divisible random variables. For Lévy processes with bounded variation sample paths, we characterize, in terms of subadditivity and concavity of the transformed Lévy tails, Lévy 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 Lévy processes without a Gaussian component, we show that the symmetric 1-stable Lévy 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 Lévy measures of multidimensional Lévy processes exhibiting the two adverse diversification phenomena uniformly over all time horizons. 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 extends beyond Lévy processes to running maxima and integrals of increasing convex functionals of Lévy processes, while both adverse diversification phenomena are preserved for Lévy-driven stochastic integrals with nonnegative deterministic kernels. Applications to ruin theory, storage processes and stochastic volatility are also discussed.
Related
- Directional Optimal Sub-Gamma Scales for Infinitely Divisible Laws
- Linear Preservers of Infinitely Divisible Matrices
- Optimal Sub-Gamma Scales for Spectrally Positive Infinitely Divisible Laws
- Minimal Radial Sub-Gamma Envelopes for Infinitely Divisible Random Vectors
- On an infinitely divisible distribution involving modified Bessel functions
BibTeX
@misc{liu2026risk,
title = {{Risk diversification for infinitely divisible distributions}},
author = {Peng Liu and Tiantian Mao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25452},
eprint = {2609.25452},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25452},
}
RIS
TY - UNPB
TI - Risk diversification for infinitely divisible distributions
AU - Peng Liu
AU - Tiantian Mao
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.25452
UR - https://arxiv.org/abs/2609.25452
AB - In this paper, we study the diversification properties of convex combinations of iid infinitely divisible random variables. For Lévy processes with bounded variation sample paths, we characterize, in terms of subadditivity and concavity of the transformed Lévy tails, Lévy 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 Lévy processes without a Gaussian component, we show that the symmetric 1-stable Lévy 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 Lévy measures of multidimensional Lévy processes exhibiting the two adverse diversification phenomena uniformly over all time horizons. 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 extends beyond Lévy processes to running maxima and integrals of increasing convex functionals of Lévy processes, while both adverse diversification phenomena are preserved for Lévy-driven stochastic integrals with nonnegative deterministic kernels. Applications to ruin theory, storage processes and stochastic volatility are also discussed.
ER -