二维经典风险模型的最优分红问题

The Optimal Dividend Problem in a Two-Dimensional Classical Risk Model

摘要: 最优分红问题是保险风险理论与公司金融问题的经典问题。由于经典风险模型的索赔额分布是任意的，为求解二维经典风险模型的值函数带来了困难。本文研究了二维经典风险模型的经典分红问题，借助逐段决定马尔可夫过程框架下盈余过程的特殊结构，通过引入优化问题：“最大化带有终端报酬的直到第一个索赔到达时刻的累计折现分红的期望”。定义了值迭代函数，证明二维经典风险模型的最优值函数可以由递增的零初值迭代函数逼近，研究了值迭代函数的最优策略，通过引入辅助优化问题，完整求解了二维经典风险模型的最优分红问题。

Abstract: The optimal dividend problem is a classic issue in insurance risk theory and corporate finance. Compared with diffusion models, the claim size distribution in the classical risk model is arbitrary, which brings difficulties to solving the value function of the two-dimensional classical risk model. This paper studies the classical dividend problem of the two-dimensional classical risk model. By leveraging the special structure of the surplus process under the PDMP framework and introducing a sub-optimization problem of the original optimal dividend problem: “maximizing the expected cumulative discounted dividends until the first claim arrival time with terminal rewards”, the expression of the iterative value function is established. It is proved that the optimal value function of the two-dimensional classical risk model can be approximated by an increasing zero initial value iterative value function. The optimal strategy of the iterative value function is studied, and the optimal dividend problem of the two-dimensional classical risk model is completely solved by introducing an auxiliary optimization problem.