一类带扰动和分红策略的相依风险模型

A Class of Dependent Risk Models with Disturbance and Dividend Strategy

摘要: 本文研究受布朗运动扰动和阈值分红策略影响的相依风险模型，即盈余低于阈值时不分红，超过阈值时以恒定速率连续支付分红，同时在模型中引入索赔阈值，规定实际索赔支付为索赔额和索赔阈值的小者。本文以Gerber-Shiu惩罚函数为研究对象，运用全概率公式、泰勒展开、布朗运动性质及极限转化等方法，推导了该风险模型的Gerber-Shiu惩罚函数满足的积分–微分方程。研究结果丰富了相依风险模型的理论体系，为保险业务的风险精确度量、分红策略优化设计提供了重要理论支撑，进一步拓展了相关模型在保险实务中的应用场景。

Abstract: This paper studies a dependent risk model influenced by Brownian motion perturbations and a threshold dividend strategy, where dividends are not paid when surplus is below the threshold and continuously paid at a constant rate when surplus exceeds the threshold. Additionally, a claim threshold is introduced into the model, stipulating that the actual claim payment is the lesser of the claim amount and the claim threshold. This paper focuses on the Gerber-Shiu penalty function and employs methods such as the total probability formula, Taylor expansion, properties of Brownian motion, and limit transformation to derive the integral-differential equations of the Gerber-Shiu penalty function under both dividend-paying and non-dividend-paying conditions. The research results enrich the theoretical system of dependent risk models, provide important theoretical support for precise risk measurement and optimal design of dividend strategies in insurance business, and further expand the application scenarios of related models in insurance practice.