CAREER: Catastrophic Rare Events: Theory of Heavy Tails and Applications

该教师早期职业发展计划（CAREER）拨款将通过开发数学工具来促进国家繁荣和福利，这些工具提供了理解和减轻与“重尾”现象相关风险的策略。 重尾分布为看似完全不同的罕见事件提供了有用的数学模型，例如全球大流行、2012年印度大停电和2007年金融危机。除了这些孤立的灾难性事件之外，重尾在大规模复杂系统和现代算法中也很普遍。重尾的一个特别简单和众所周知的表现是所谓的“80-20规则”，其变化在各种各样的应用领域中被反复发现。在存在重尾的情况下，高影响力的罕见事件最终肯定会发生，而且发生的频率可能比决策者可能考虑的要高。 考虑（甚至利用）这种罕见事件造成的影响将支持在许多重要场景中设计和运行可靠和有弹性的系统，包括环境灾难，电力系统故障，金融危机。 伴随的教育计划旨在扩大STEM在代表性不足的社区的兴趣，并通过装备他们在风险分析的基本技能，培养学术界，工业界和政府的未来领导人。这项研究将开发一个全面的理论的大偏差和亚稳定性的重尾随机系统。经典的大偏差理论和稀有事件模拟有着悠久的历史，但当潜在的不确定性是重尾的时，这些方法和亚稳定性框架往往不足。 该项目利用并扩展了极值理论，优化，控制和随机模拟的最新进展，通过构建为重尾系统量身定制的大偏差和亚稳态框架来填补差距。 通过新的框架，该项目还将解决人工智能和精算科学中的开放问题。 这项研究将为设计可靠和负责任的人工智能提供严格的理论基础，以便该技术可以应用于高风险决策问题。 该项目的成功实施将扩大我们对许多随机系统中如何出现系统故障和相变的理解，从而为保险风险管理和负责任的人工智能设计提供可证明的高效计算机制。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This Faculty Early Career Development Program (CAREER) grant will contribute to the advancement of national prosperity and welfare by developing mathematical tools that provide strategies to understand and mitigate risk associated with the "heavy-tail" phenomena. Heavy-tailed distributions provide useful mathematical models for seemingly disparate rare events, such as the global pandemic, the 2012 blackout in India, and the 2007 financial crisis. Beyond such isolated catastrophic events, heavy tails are pervasive in large-scale complex systems and modern algorithms. A particularly simple and well-known manifestation of heavy tails is the so-called “80-20 rule”, whose variations are repeatedly discovered in a wide variety of application areas. Under the presence of heavy tails, high-impact rare events are guaranteed to happen eventually, and may occur more frequently than decision-makers may account for. Accounting for (or even utilizing) the impact inflicted by such rare events will support the design and operation of reliable and resilient systems in many important scenarios, including environmental catastrophes, power system failures, financial crises. The accompanying educational plan aims to broaden STEM interest in underrepresented communities and train future leaders of academia, industry, and government by equipping them with fundamental skills in risk analysis.This research will develop a comprehensive theory of large deviations and metastability for heavy-tailed stochastic systems. The classical theory of large deviations and rare-event simulation has a long history but these approaches and the metastability framework often fall short when the underlying uncertainties are heavy-tailed. This project leverages and extends recent advances in extreme value theory, optimization, control, and stochastic simulation to fill the gap by building large deviations and metastability frameworks tailored for heavy-tailed systems. With the new framework, the project will also address open problems in artificial intelligence and actuarial science. This research will contribute to a rigorous theoretical foundation for designing reliable and accountable AI so that the technology can be applied to high-stake decision-making problems. Successful implementation of such a program will expand our understanding of how system failures and phase transitions arise in many stochastic systems, which, in turn, will provide provably efficient computational machinery for insurance risk management and accountable AI design.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.