具有分红和随机扰动相依模型的Gerber-Shiu惩罚函数

Gerber-Shiu Penalty Function with Dividend and Random Disturbance Dependent Model

摘要: 本文研究一类受布朗运动扰动的相依风险模型，并在模型中考虑分红策略，以Gerber-Shiu惩罚函数为研究对象，首先得到了其在有分红和无分红两种情况下的积分–微分方程，然后研究无红利支付与阈值分红策略下的Gerber-Shiu惩罚函数，最后基于矩阵商差理论和线性独立解构造，推导出有分红情形下Gerber-Shiu惩罚函数的显式表达式。研究结果为保险风险的精确度量与分红策略的优化设计提供了理论支撑，拓展了经典风险模型的应用场景。

Abstract: This paper investigates a class of dependent risk models perturbed by Brownian motion, incorporating dividend strategies within the model. Focusing on the Gerber-Shiu penalty function, we first derive its integro-differential equations under both dividend-paying and non-dividend-paying scenarios. Subsequently, we explore the Gerber-Shiu penalty function under non-dividend payment and threshold dividend strategies. Lastly, based on matrix quotient difference theory and linear independent solution construction, we deduce an explicit expression for the Gerber-Shiu penalty function in the dividend-paying case. The research findings provide theoretical support for the precise measurement of insurance risks and the optimal design of dividend strategies, expanding the application scenarios of classical risk models.