Model uncertainty and robustness in risk management

本提案的总体目标是发展金融和保险风险管理中模型不确定性和鲁棒性的数学和经济理论以及统计和计算方法。模型不确定性和稳健性问题，涉及“不正确、不合理或滥用的模型输出和报告”，已成为当前风险管理和监管挑战的核心组成部分。我们沿着几个在金融和保险中具有实际重要性的方向解决模型不确定性。首先，由于高维统计建模的挑战和数据的限制，风险聚合中的依赖不确定性在实践中是一个具有挑战性但又很常见的情况，是近年来风险管理中一个非常热门的研究课题。众所周知，开发一种风险评估程序，使其能够计算出依赖不确定性下实际可雇用的风险汇总评估，是一项极具挑战性的工作。我们通过借鉴鲁棒优化和依赖建模文献中最新研究发展的技术来解决这个问题。其次，我们研究了风险度量和风险优化中的鲁棒性问题，从而解决了根据稍微错误的假设（最相关的情况）管理和优化风险的定量后果。第三，我们将模型不确定性引入到风险分担和经济均衡问题中，并分析了模型不确定性和异质信念在复杂金融系统中的影响。在此过程中，我们为与上述问题相关的领域开发了深刻的数学工具，例如度量理论，决策理论，copuls，博弈论，统计鲁棒性，非凸优化和概率组合。

The overall aim of this proposal is to develop the mathematical and economic theory as well as statistical and computational methods for model uncertainty and robustness in financial and insurance risk management. Model uncertainty and robustness issues, concerning “incorrect, unjustified or misused model outputs and reports", have appeared as a central component of the current challenges in risk management and regulation. We address model uncertainty along several directions of practical importance in finance and insurance. First, dependence uncertainty in risk aggregation, as a challenging yet common situation in practice due to challenges in high-dimensional statistical modeling and data limitation, has recently been a very popular research topic in risk management. Developing a risk evaluation procedure which allows for computing practically employable assessments of risk aggregation under dependence uncertainty is well known to be highly challenging. We address this problem by borrowing techniques from recent research development in the literature of robust optimization and dependence modeling. Second, we investigate robustness issues in risk measures and the optimization of risk, thus addressing the quantitative consequences of managing and optimizing risks according to slightly wrong assumptions, a most relevant situation. Third, we bring model uncertainty into the problems of risk sharing and economic equilibrium for various settings of risk managers and financial contexts, and analyze the effect of model uncertainty and heterogeneous beliefs in a complex financial system. Along the way, we develop profound mathematical tools for fields related to the above problems, such as measure theory, decision theory, copulas, game theory, statistical robustness, non-convex optimizations, and probabilistic combinatorics. The proposed research will contribute significantly to the field of quantitative risk management via a comprehensive understanding of risk and uncertainty. The anticipated results will provide novel theoretical insights and will lead to advances in various practical domains, including regulatory policy making, enterprise risk management and economical decision making. The proposed mathematical and statistical developments have wide applications wherever uncertainty in a complex model is concerned, thus being useful in other fields such as operations research, management science, natural sciences, informatics, and statistics. Data analysis will also play an important role and we shall mainly work with financial data. HQP will be intensively involved in the research process. They will gain skills in conducting research and implementing data science technologies, learn knowledge in statistics, actuarial science and risk management, and fulfill the research requirements in their graduate programs through the proposed research.