Abstract: In the insurance industry, both the non-claims premium discount system and the deductible system contribute to unequal occurrences of claim for compensation and settlement of claims. MAO Zechun and LIU Jin'e (2004,2005) introduced the compound Poisson-Geometric distribution and process, also known as the Polya-Aeppli process internationally, to characterize this phenomenon. Compound Poisson-Geometric processes, as an extension of compound Poisson processes, have attracted research attention from scholars in financial mathematics, particularly focusing on mean-variance models, utility models, and bankruptcy probability models.
The dividend strategy serves as a crucial risk control method that not only stimulates policyholder engagement, but also boosts insurance companies' premium income and enhances their solvency. This approach has been widely embraced by each insurance company as part of its management strategy. While some scholars have examined the optimal dividends for insurance companies under compound Poisson-Geometric risk processes, they have not taken into account the investment of risk-free assets for the convenience of mathematical calculations. Risk-free investments are commonly utilized by insurance companies to maximize returns while ensuring high security, stable returns, low volatility, and good liquidity. Therefore, it is necessary to consider the investment of risk-free assets.
This paper examines the compound Poisson-Geometric risk model with risk-free investment under optimal portfolio-cheap reinsurance and barrier dividend. By utilizing dynamic programming principles, we obtain and solve the HJB equation while obtaining analytic solutions to optimal investment-cheap reinsurance and optimal dividend functions.
Finally, we analyze how changes in key parameters such as the risk-free interest rate influence optimal investment strategies and dividend functions,verify the rationality of our results,and propose management suggestions: from the perspective of stimulating insurance enthusiasm and increasing insurance dividends, increasing initial reserves and investing in risky assets with high yield, low volatility, low risk correlation with claims and risk-free assets with high yield are effective ways to increase dividends. At the same time, when the risk-free interest rate is high, it is also a wise strategy for insurance companies to switch from risky assets to risk-free assets. From the perspective of risk transfer, under the high yield and low volatility of risk assets, for the purpose of pursuing dividends, it is necessary to increase reinsurance, accept more risk investment, increase the transfer insurance risk and maintain the stability of the overall risk. The greater the correlation coefficient, if the volatility of risk asset is larger, the more conducive to dividends the reduction of reinsurance.

Key words: compound Poisson-Geometric process, cheap-reinsurance, barrier dividend, deviation coefficient

摘要： 文章研究了带无风险投资的最优投资组合-便宜再保-障碍分红下的复合Poisson-Geometric风险模型,通过使用动态规划原理得到并求解了HJB方程,解得最优投资-便宜再保与最优分红函数的解析解。最后分析了无风险利率等关键参数对模型结果的影响,验证了建模的合理性,并给出了经营策略。这些建议包括:从激发投保热情,增加投保人分红的角度看,增加初始准备金,投资高收益率、低波动率、且与索赔风险相关度低的风险资产和收益率高的无风险资产都是提高分红的有效途径。同时在无风险利率较高时,保险公司从风险资产转投无风险资产也不失为一种明智的策略。从转移风险的角度看,风险资产的高收益率,低波动率下,出于追求分红的目的,反而要增加再保,接受更多的风险投资,就要增加转移保险风险,维持整体风险的稳定;相关系数越大,若风险资产波动率较大,反而要减少再保更有利于分红。

关键词: 复合Poisson-Geometric过程, 便宜再保, 障碍分红, 偏离系数