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

金融危机表明，需要有复原力的金融和保险市场。因此，保险市场的一个关键考虑因素是稳健保险合同的设计。实际上，最优保险设计理论是精算学的基石之一，从保险购买者的角度来看，什么样的保险合同是最优的问题是其核心。这个问题的严格处理需要一个数学公式的最优性标准。经典理论建立在Arrow（1971）的基础上，并植根于不确定性下的经典选择模型，即期望效用理论（EUT）：保险寻求者是风险厌恶的EU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显示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，同时保持了EUT的假设，即个人是完全理性的，并且精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，模式设计是不合理的意义上的EUT和经典的保险模式是过于局限性。例如，在模糊（模型不确定性）的情况下，旅游目的地管理者无法充分评估所涉的概率环境，以及旅游目的地管理者在评估可能性方面不同于保险公司的情况。对新兴风险的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行为模型至关重要，以便使理论预测与现实相一致，适当地为政策制定和保险市场监管提供信息，并指导精算实践和有效的合同设计。这是本文提出的研究计划的长期目标，我以前的研究的核心组成部分集中在推进这一努力。建议的研究计划将继续在这条道路上建立在我以前的工作，并将信念异质性，模糊性厌恶，损失厌恶最优保险设计。** 在技术层面上，这项研究将提出严重的数学挑战，这些挑战来自贝叶斯优化和/或经典测量理论方法在模糊性和/或损失规避环境中的不适用性。最优保险设计问题将被表述为涉及非可加概率测度和Choquet积分的非凸优化问题。新的技术需要基于非加性测量理论，本文提出的研究将利用我以前的工作来实现这一点。正如我的研究计划中详尽描述的那样，学生培训将嵌入到这个研究计划的每个阶段。

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.