Collaborative Research: Modeling and Analyzing Extreme Risks in Insurance and Finance

最近发生的具有灾难性经济和社会后果的罕见事件，即所谓的黑天鹅事件，使今天的世界与几十年前大不相同。这些事件的例子很多，从地震和核危机，到次级抵押贷款引发的金融市场崩溃。所有这些都加剧了保险和金融行业对风险管理的需求，特别是对极端风险进行建模、分析、预测和管理的新工具的发明。研究人员将承担罕见事件研究带来的根本挑战：他们本质上缺乏数据，他们的可能性很难计算，而且他们很难通过准确的模型反映出来。因此，调查人员将采取综合方法，结合统计方法、概率分析、优化和有效的计算机模拟来评估其风险。考虑到项目中记录的模型和方法在处理极端事件的重要影响方面的广泛应用，这项研究具有很高的社会影响。调查人员计划培训几名博士生，并通过与哥伦比亚大学的哈莱姆学校合作伙伴关系，让他们作为客座讲师参与K12教育。调查人员将试图从代表性不足的群体中招聘高素质的人员。他们还将通过开放获取网站传播该项目的科学成果。该项目的智力优势在于它包括算法、计算、统计和理论部分。目标是建立一种稳健和系统的方法来建模和分析保险和金融中的极端风险。研究人员系统地结合了：a)极值统计理论，b)概率中的渐近分析，c)随机优化，d)高效的蒙特卡罗技术。具体目标包括：1)建立一个稳健的渐近理论，以在广泛的设置范围内一致地解释尾部事件；2)利用渐近大偏差理论，为罕见事件建立被证明有效的蒙特卡罗估计量；以及3)研究一种稳健的优化方法，以解决模型的错误指定。研究人员的方法既是创新的，也是必要的，因为罕见事件的性质暴露了这些领域中的每个方面的缺陷：i)极值统计理论假设了大量数据点，以提供准确的估计值，但罕见事件的性质排除了这一假设。Ii)渐近分析技术允许获得易于评估的公式，从而能够在大范围的模型参数下进行灵敏度分析。因此，渐近可能有助于缓解一些统计误差问题，但不幸的是，它们带来了无法测量的误差，往往会丢失重要信息。Iii)有效的蒙特卡罗通过抽样具有可控的误差，但它仍然假设模型是适当的，如果模型不正确，可能会导致错误的结论。Iv)优化技术可以通过计算感兴趣的概率的界限，以高置信度优化覆盖真相的模型的选择，来帮助缓解模型不确定性的问题。然而，这些可能太松散而不实用，或者优化问题可能太复杂，因此需要从1)到3)。总体而言，调查人员将建立一种全面的办法，解决上述每一种隔离方法所暴露的缺陷。

Recent rare events with disastrous economic and social consequences, so-called Black-Swan events, have made today's world far different from just decades ago. Examples of these events range from earthquakes and nuclear crises to the collapse of financial market from sub-prime mortgages. All of these intensify the need for risk management among the insurance and financial industry, and in particular, the invention of new tools to model, analyze, predict, and manage extreme risks. The investigators will undertake the fundamental challenges posed by the study of rare events: they, by nature, are short of data, their likelihood is difficult to compute, and that they are difficult to reflect via accurate models. As such, the investigators will take an integrated approach that combines statistical methods, probabilistic analysis, optimization, and efficient computer simulation to assess their risks. The research has potential for high societal impact, given the wide range of applications of the models and methods documented in the project for handling the important implications of extreme events. The investigators plan to train several Ph.D. students and will also involve them in K12 education as guest lecturers via Harlem Schools Partnership with Columbia University. The investigators will attempt to recruit high-quality personnel from under-represented groups. They will also disseminate the scientific output of this project via open access sites.The intellectual strength of the project rests on the fact that it includes algorithmic, computational, statistical, and theoretical components. The goal is to establish a robust and systematic approach for modeling and analyzing extreme risks in insurance and finance. The investigators systematically combine: a) the theory of extreme value statistics, b) asymptotic analysis in probability, c) stochastic optimization, and d) highly efficient Monte Carlo techniques. Specific objectives include: 1) establishing a robust asymptotic theory to account for tail events uniformly over a wide range of settings, 2) taking advantage of the asymptotic large deviations theory to build provably efficient Monte Carlo estimators for rare events, and 3) investigation of a robust optimization approach that accounts for model misspecification. The investigators' approach is both innovative and necessary because the nature of rare events exposes deficiencies in each of these areas: i) The theory of extreme value statistics assumes a large number of data points to provide accurate estimators but the nature of rare events precludes this assumption. ii) Asymptotic analysis techniques allow to obtain formulas that are easily evaluated and thus amenable to sensitivity analysis under a wide range of model parameters. So, asymptotics may help mitigate some of the statistical error issue, but unfortunately, they carry an unmeasurable error and often lose important information. iii) Efficient Monte Carlo has a controlled error by sampling, but it still assumes a model in place and could lead to incorrect conclusions in case the model is incorrect. iv) Optimization techniques might help mitigate the problem of model uncertainty by computing bounds for the probabilities of interest, optimizing over the selection of models that cover the truth with high confidence. However, these might be too loose to be practical or the optimization problem could be too complex, thus the need from 1) to 3). Overall, the investigators will establish a comprehensive approach that resolves the deficiencies exposed by each of the aforementioned segregated methods.