Path-dependent measures of risks: drawdowns, occupation times and Parisian times

与2007-2008年的金融危机一样，我们经历了一个完整的繁荣与萧条周期：危机前的经济泡沫，市场的突然崩溃，以及随后的衰退。金融危机不仅对全球经济造成了巨大的破坏，而且对保险和金融领域量化风险管理的实践产生了广泛的影响。它促使加拿大和世界各地的研究人员开发和改进数学模型，以更好地表征和衡量这些极端风险。**拟议的研究将系统地研究缩减、职业时间和巴黎时间（DOP），它们作为表征和衡量崩溃和萧条极端风险的自然模型。由于路径依赖的特点，DOP能够提供一个时间段内保险业务或投资组合业绩的更可靠、更全面的信息。然后，风险管理决策可以及时做出，这是至关重要的。此外，最近的许多研究表明，DOP在对保险和金融中的各种事件、工具和法规进行建模方面具有广泛的应用。**本研究的目标是：(1)发展DOP理论分析和数值计算的基本理论和一般方法；(2)更好地理解崩盘和萧条等极端风险；(3)完善量化风险管理技术，衡量和降低极端风险。计划中的研究每年将产生2-3篇单独或联合论文，发表在精算科学、应用概率或数学金融的顶级期刊上。研究生和优秀的本科生将集中参与这项拟议的研究计划。

As in the 2007-2008 financial crisis, we have experienced a complete cycle of booms and busts: the economic bubble before the crisis, the sudden crash of the market, as well as the recession thereafter. The financial crisis not only caused huge damage to the global economy but also had extensive impact on the practice of quantitative risk management in insurance and finance. It prompts researchers in Canada and all over the world to develop and improve mathematical models to better characterize and measure these extreme risks. **The proposed research will systematically study drawdowns, occupation times, and Parisian times (DOP) which serve as natural models to characterize and measure the extreme risks of crash and depression. Because of the features of path dependency, DOP are able to provide with more reliable and comprehensive information on the performance of insurance businesses or investment portfolios in a time period. Then, risk management decisions can be taken in a timely manner which is of paramount importance. Moreover, much of recent research has showed that DOP have wide applications in modeling various events, instruments and regulations in insurance and finance.**The objectives of this proposed research are (1) to develop fundamental theory and general methodology in both theoretical analysis and numerical computation of DOP; (2) to pursue a better understanding of extreme risks such as crash and depression; (3) to improve quantitative risk management techniques to measure and reduce the extreme risks. The proposed research will produce 2-3 sole or joint papers per year to be published in the top tier journals in actuarial science, applied probability or mathematical finance. Graduate students and exceptional undergraduate students will be intensively involved in this proposed research program.