We consider the problem of minimizing the probability of ruin by purchasing reinsurance whose premium is computed according to the mean-variance premium principle, a combination of the expected-value and variance premium principles. We derive closed-form expressions of the optimal reinsurance strategy and the corresponding minimum probability of ruin under the diffusion approximation of the classical Cramér-Lundberg risk process perturbed by a diffusion. We find an explicit expression for the reinsurance strategy that maximizes the adjustment coefficient for the classical risk process perturbed by a diffusion. Also, for this risk process, we use stochastic Perron's method to prove that the minimum probability of ruin is the unique viscosity solution of its Hamilton-Jacobi-Bellman equation with appropriate boundary conditions. Finally, we prove that, under an appropriate scaling of the classical risk process, the minimum probability of ruin converges to the minimum probability of ruin under the diffusion approximation.
A Stackelberg reinsurance-investment game under $α$-maxmin mean-variance criterion and stochastic volatility
本文研究了在$α$ -maxmin均值-方差准则下保险人和再保险人之间的Stackelberg博弈。保险人可以向再保险人购买按损失再保险。在Stackelberg博弈中，再保险人采用反馈再保险策略优化再保险保费。金融市场由现金和股票组成，具有赫斯顿随机波动率。保险人和再保险人都最大化各自的$α$ -maxmin均值-方差偏好。该准则是时间不一致的，我们利用扩展的哈密顿-雅可比-贝尔曼方程导出了平衡策略.与Li and Young（2022）的非稳健情形类似，超额损失再保险是保险公司的最优再保险策略。均衡投资策略由一个Riccati微分方程组确定。此外，半显式地给出了确定均衡再保险策略和再保险费率的方程，并在具体算例中将其简化为一个代数方程。数值算例表明，保险人和再保险人之间的博弈使得保险更加激进，当代理人变得更加模糊厌恶或风险厌恶。此外，模糊性水平、模糊态度和保险人（再保险人）的风险态度对均衡再保险策略、再保险保费和投资策略也有类似的影响。
Optimal Reinsurance-Dividend Strategy with Fixed Transaction Costs in a Regime-Switching Brownian Risk Model: A Viscosity Solution to the Impulse Control Problem
Fujian Normal University（福建师范大学）；The Hong Kong Polytechnic University（香港理工大学）；Xi’an Jiaotong University（西安交通大学）
