Optimal (Re)Insurance Design: Ambiguity, Belief Heterogeneity, and Loss Aversion

金融危机表明，需要有弹性的金融和保险市场。因此，保险市场的一个关键考虑因素是设计稳健的保险合同。事实上，最优保险设计理论是精算科学的基石之一，从保险买家的角度来看，什么保险合同是最优的是它的核心问题。对这个问题的严格处理需要一个最优标准的数学公式。经典理论建立在Arrow(1971)奠定的基础之上，并植根于不确定条件下的经典选择模型，即期望效用理论(EUT)：保险寻求者是风险厌恶的欧盟最大化决策者(DM)，面临由给定概率空间上的随机变量表示的可保损失。在这种情况下，经典的贝叶斯优化方法可以用来证明线性免赔额合同的最优性。这些基本结果在几个方向上得到了扩展，同时维持了EUT的假设，即个人是完全理性的，并准确地知道与任何决策情况相关的可能性。然而，有大量的经验证据表明，DM在EUT意义上是不合理的，经典的保险模型过于有限。例如，在不明确的情况下(模型不确定)，DM不能充分评估所涉及的概率环境，以及DM在评估可能性方面与保险公司不同的情况。新兴风险保险就是一个很好的例子。在最优保险设计问题中，构建更现实的DM行为模型是至关重要的，这样才能使理论预测与实际相符，为政策制定和保险市场监管提供适当的信息，并指导精算实践和有效的合同设计。这是这里提出的研究计划的长期目标，我之前研究的一个核心部分集中在推进这一努力上。建议的研究计划将继续沿着这条道路前进，在我之前工作的基础上，将信念异质性、歧义厌恶和损失厌恶纳入最优保险设计。在技术层面上，这项研究将带来严重的数学挑战，因为贝叶斯优化和/或经典测量理论方法在模糊性和/或损失厌恶的背景下不适用。最优保险设计问题将被描述为涉及非加性概率度量和Choquite积分的非凸优化问题。需要基于非加性测量理论的新技术，这里提出的研究将利用我之前的工作来实现这一点。正如我在研究计划书中详细描述的那样，在每个阶段，学生培训都将包含在这个研究计划中。

Financial crises have demonstrated the need for resilient financial and insurance markets. A key consideration in insurance markets is then the design of robust insurance contracts. Indeed, the theory of optimal insurance design is one of the cornerstones of actuarial science, and the question of what insurance contract is optimal from an insurance buyer's perspective lies at its core. A rigorous treatment of this question requires a mathematical formulation of an optimality criterion. The classical theory builds upon the foundations laid out by Arrow (1971) and is rooted in the classical model of choice under uncertainty, i.e. Expected-Utility Theory (EUT): an insurance seeker is a risk-averse EU-maximizing decision-maker (DM) facing an insurable loss represented by a random variable on a given probability space. In this case, classical Bayesian optimization methods can be used to show the optimality of a linear deductible contract. These foundational results have been extended in several directions while maintaining EUT's assumption that individuals are fully rational and know precisely the likelihoods associated with any decision-making situation. However, there is substantial empirical evidence that DMs are not rational in the sense of EUT and that the classical insurance model is too limiting. For instance, there are situations of ambiguity (model uncertainty) in which DMs are not able to fully assess the probabilistic environment involved, as well as situations where DMs differ from insurers in their assessment of likelihoods. Insurance of emerging risks is a prime example.It is critically important to construct more realistic models of DM behaviour in problems of optimal insurance design so as to align theoretical predictions with reality, suitably inform policy-making and insurance market regulation, and guide actuarial practice and efficient contract design. This is the long-term goal of the research program proposed herein, and a core component of my previous research has focused on advancing this effort. The proposed research program will continue on this path by building upon my previous work and incorporating belief heterogeneity, ambiguity aversion, and loss aversion in optimal insurance design. On a technical level, this research will present serious mathematical challenges arising from the inapplicability of Bayesian optimization and/or classical measure-theoretic methods in a setting of ambiguity and/or loss aversion. The optimal insurance design problems will be formulated as non-convex optimization problems involving non-additive probability measures and Choquet integration. Novel techniques are needed based on non-additive measure theory, and the research proposed herein will leverage my previous work to accomplish this. Student training will be embedded in this research program at every stage, as exhaustively described in my research proposal.