Paper: arXiv 2609.25452

Authors: Peng Liu, Tiantian Mao

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.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper is highly theoretical, focusing on advanced mathematical characterizations of diversification phenomena in infinitely divisible distributions and Lévy processes. While it introduces novel insights into non-diversification, it lacks empirical validation or backtesting, placing it firmly in the ‘Lab Rats’ quadrant. The mathematical derivations are complex and central to its contribution.

Research Flowchart

  flowchart TD
    A[Research Goal: Study Diversification Properties of Convex Combinations of i.i.d. Infinitely Divisible RVs] --> B{Key Methodology Steps};

    B --> C1[Characterize Subadditivity/Concavity of Transformed Lévy Tails for Bounded Variation Lévy Processes];
    B --> C2[Analyze Symmetric Lévy Processes without Gaussian Component];
    B --> C3[Investigate Convex Combinations of Multidimensional Infinitely Divisible Distributions];
    B --> C4[Extend Phenomena to Running Maxima, Integrals, and Stochastic Integrals of Lévy Processes];

    C1 & C2 & C3 & C4 --> D[Data/Inputs: Infinitely Divisible Random Variables, Lévy Processes, Multidimensional Distributions];

    D --> E[Computational Processes: Mathematical Analysis, Characterizations, Proofs];

    E --> F[Key Findings/Outcomes: Characterizations of Non-diversification & Reverse Diversification for various Lévy process types, Extension to related stochastic processes, Applications in Ruin Theory, Storage, Volatility];