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

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