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September 10, 2025

Joint Tail Probability of Renewal Models of Dependent Heavy‐Tailed Random Variables With Applications to Systemic Risk Measures

LZLei ZouTexas A&M University JPJiangyan PengUniversity of Electronic Science and Technology of China CXChenghao XuUniversity of Electronic Science and Technology of China

Key Points

Derived asymptotic formulas for joint tail probabilities of aggregate discounted claims reveal crucial dependencies.
The analysis utilizes a Lévy process to model claims arriving in pairs, highlighting heavy-tailed behavior.
Numerical studies illustrate theoretical results, indicating implications for evaluating systemic risk measures.
This work emphasizes the significance of tail dependence and marginal tail characteristics in risk assessment.

Abstract

ABSTRACT Consider a non‐standard renewal risk model in which claims arrive in pairs and the stochastic discounting process is given by , where is a Lévy process. We are interested in the joint tail probability of and , the aggregate discounted claims along the respective lines . Assume that is a sequence of independent and identically distributed random pairs with generic pair . Further assume that follows a dependence structure encompassing both tail dependence and tail independence, and has regularly varying marginal tails. We derive asymptotic formulas for the joint tail probability of and . These results are then applied to evaluate two systemic risk measures. Finally, we conduct numerical studies to illustrate the theoretical findings.

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