无限可分分布的风险分散化
Risk diversification for infinitely divisible distributions
- School of Mathematics, Statistics and Actuarial Science, University of Essex（埃塞克斯大学数学、统计与精算科学学院）
- Department of Statistics and Finance, School of Management, University of Science and Technology of China（中国科学技术大学管理学院统计与金融系）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文刻画了无限可分分布凸组合的分散化性质，证明对称1-稳定Lévy过程是唯一无分散化效应的对称过程，并将结果推广至多维及路径依赖过程。
AI中文摘要：
本文研究了具有无限可分分布的独立同分布随机变量凸组合的分散化性质。我们通过变换后的Lévy测度的次可加性和凹性，刻画了在所有时间范围内关于优序一致表现出非分散化现象或反向分散化顺序的Lévy过程。特别地，我们证明了对称1-稳定Lévy过程是唯一表现出非分散化现象的对称Lévy过程。我们进一步研究了多元无限可分分布各分量的凸组合，允许风险具有依赖性和异质性分布，并刻画了相应多维Lévy过程随时间一致表现出这两种分散化现象的Lévy测度。对于多维对称Lévy过程、多维α-稳定过程和多维复合泊松过程，我们获得了显式刻画。最后，我们证明非分散化现象和反向分散化顺序不仅限于Lévy过程，还扩展到若干样本路径依赖过程，包括运行最大值和Lévy驱动的随机积分，其中Lévy驱动的Ornstein-Uhlenbeck过程是一个重要的特例。此外，还讨论了在破产理论、存储过程和随机波动率中的应用。
英文摘要：
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.
