狭窄框架下保险模型的Stackelberg博弈问题

A Stackelberg Game Problem in Insurance Models with Narrow Framing

摘要: 考虑狭窄框架和均值-方差准则下的 Stackelberg 博弈再保险问题。 狭窄框架意昧着，购买保险的 动机除了对冲财富风险，还有可能将购买保险本身看作一项风险投资。 因此，使用二次效用函数 来度量保险净收益的局部得失效用，即，狭窄框架。 在 Stackelberg 博弈中，再保险公司首先向 保险公司提供合理的赔偿来换取适当的保费。 然后，保险公司根据这个保费原则选择最优的赔偿。 首先，假设再保险公司选定期望值保费原则，保险公司通过选择最优赔偿策略来最大化终端财富 的均值-方差函数和保险净收益的二次函数。 然后给定这个最优赔偿，再保险公司通过选择最优保 费参数最大化终端财富的均值-方差函数。 此外，考虑另外一个 Stackelberg 博弈的问题。 对于保 险公司来说，考虑与前者相同的目标函数，给定Ⅱ(I) = E(PI) 保费原则，得到了最优保险赔偿的 表达式。 之后，给定这个最优的保险赔偿，最大化再保险公司终端财富的期望效用准则，计算得 到了保费的最优价格强度。 进一步，通过 Taylor 展开，得到了这对最优解的近似表达式。

Abstract: In this paper, we consider the Stackelberg game reinsurance problem under mean- variance criterion with narrow framing. The motivation for purchasing insurance might not only be hedging wealth risk, but also to consider the purchase of insurance itself as a risky investment, which is called narrow framing. Inspired by this, we use a quadratic utility function to measure the local gain-loss utility of the net benefits of insurance, namely, narrow framing. As the Stackelberg game in insurance models, the reinsurer first offers the insurer a reasonable indemnity in exchange for the appropri- ate premium. Then, the insurer selects an indemnity based on the premium principle. In this paper, suppose that the reinsurer chooses the expected value premium prin- ciple, we compute the optimal insurance indemnity to maximize the mean-variance functional of the insurer’s terminal wealth and the quadratic function of net insurance returns. Then, given the optimal indemnity of the insurer, we compute the parameter of the expected value premium principle to maximize the mean-variance function of the reinsurer’s terminal wealth. In addition, we consider another Stackelberg game. For the insurer, by considering the same objective function as the former and giving the premium principle Ⅱ(I) = E(PI), we obtain an expression of the optimal indem- nity. Afterwards, given the optimal indemnity, we maximize the expected utility for the terminal wealth of the reinsurer and compute the optimal price intensity of the premium. Furthermore, by applying Taylor expansion, we find the approximation expression of the optimal solution.