In this paper, we consider a dynamic Pareto optimal risk‐sharing problem under the time‐consistent mean‐variance criterion. A group of n insurers is assumed to share an exogenous risk whose dynamics is modeled by a Lévy process. By solving the extended Hamilton–Jacobi–Bellman equation using the Lagrange multiplier method, an explicit form of the time‐consistent equilibrium risk‐bearing strategy for each insurer is obtained. We show that equilibrium risk‐bearing strategies are mixtures of two common risk‐sharing arrangements, namely, the proportional and stop‐loss strategies. Their explicit forms allow us to thoroughly examine the analytic properties of the equilibrium risk‐bearing strategies. We later consider two extensions to the original model by introducing a set of financial investment opportunities and allowing for insurers' ambiguity towards the exogenous risk distribution. We again explicitly solve for the equilibrium risk‐bearing strategies and further examine the impact of the extension component (investment or ambiguity) on these strategies. Finally, we consider an application of our results in the classical risk‐sharing problem of a pure exchange economy.

中文翻译：

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时间一致均值-方差准则下的最优动态风险分担

在本文中，我们考虑了时间一致的均值-方差准则下的动态帕累托最优风险分担问题。一组n假设保险公司分担外部风险，其动态由Lévy流程建模。通过使用拉格朗日乘数法求解扩展的汉密尔顿-雅各比-贝尔曼方程，可以获得每个保险公司时间稳定均衡风险承担策略的显式形式。我们证明均衡的风险承担策略是两种常见的风险共享策略的混合，即比例策略和止损策略。它们的显式形式使我们能够彻底检查均衡风险承担策略的分析性质。稍后，我们通过引入一组金融投资机会并考虑到保险公司对外部风险分配的含糊不清，考虑了对原始模型的两个扩展。我们再次明确地解决了平衡风险承担策略，并进一步研究了扩展成分（投资或歧义性）对这些策略的影响。最后，我们考虑将结果应用于纯交换经济的经典风险分担问题中。