CAREER: Bridging High-Frequency Data Analysis and Continuous-time Features of Levy Models

受该领域最新理论发现的启发，研究人员确定了 Levy 过程在短时间内渐近行为的一些关键开放问题，并将它们与应用中常见的两个重要统计问题联系起来：Levy 模型的参数估计和变点检测。 对于有限随机样本，已知最大似然估计和累积和 (CUSUM) 顺序规则等方法是处理前面提到的两个问题的最佳方法。尽管人们期望当 Levy 过程的连续观察之间的时间跨度缩小到零时，最优性将得以保留，但有重要的例子表明情况并非总是如此。这些违反直觉的结果背后的奥秘与 Levy 过程在短时间内的“精细”分布特性密切相关。研究人员不是直接解决连续时间内提出的两个问题，而是建立在离散时间上经过充分研究的类似问题的基础上，并通过分析当连续观察之间的时间跨度越来越小时它们的演化来填充无限时间连续体。这种自下而上的方法不仅有吸引力，而且很有用，因为在实践中，人们希望确定高频观测而不是连续时间观测的统计方法的性能，而连续时间观测可以说是永远无法实现的。对 Levy 过程的关注是因为后者是最简单的随机模型，显示突然的变化，同时仍然保留其增量的简约统计特性。还考虑扩展到由 Levy 过程驱动的其他多因素随机模型。自然和社会现象的自动高频监测系统越来越多地应用于工程应用、金融市场和环境研究。因此，对这些系统生成的高频数据的高效、准确的统计和计算方法的需求日益增长。这种需求产生了两个重要问题：理解高频采样设置中统计效率的含义，以及分析一些常用统计方法应用于高频数据时的最优性。所进行的研究回应了这两个紧迫的问题。该项目的成果在金融衍生品定价、金融模型校准、导航系统监控、计算机网络入侵检测等方面具有重要应用。教育影响包括为本科生提供暑期研究经验，以及为统计、概率和数学金融等跨学科主题开发教学/计算资源。这些活动涉及研究生，并针对科学领域代表性不足的群体的参与。

Motivated by recent theoretical findings in the field, the investigator identifies some key open problems of the asymptotic behavior of Levy processes in short time and connects them to two important statistical problems commonly appearing in applications: parametric estimation and change-point detection for Levy models. For finite random samples, methods such as maximum likelihood estimation and cumulative sum (CUSUM) sequential rules are known to be optimal for dealing with the two previously mentioned problems. Although one expects that optimality would be preserved when the time span between consecutive observations of a Levy process shrinks to zero, there exist important examples showing this not always to be the case. The mystery behind these counterintuitive results is closely connected to the "fine" distributional properties of Levy processes in short time. Rather than directly attacking the two proposed problems in continuous time, the investigator builds on the well-studied analogous problems in discrete time and fill in the infinite time continuum by analyzing their evolution when the time span between consecutive observations is made increasingly small. This bottom-up approach is not only appealing but also useful since in practice one would like to determine the performance of statistical methods for high-frequency observations rather than for continuous-time observations, which are arguably never available. The focus on Levy processes is motivated by the fact that the latter are the simplest stochastic models displaying abrupt changes while still preserving the parsimonious statistical properties of their increments. Extensions to other multi-factor stochastic models driven by Levy processes are also contemplated. Automatic high-frequency monitoring systems of natural and social phenomena are increasingly used in engineering applications, financial markets, and environmental studies. Therefore, there is an increasing need for efficient and accurate statistical and computational methods for the high-frequency data generated by these systems. Two important issues arise with this need: understanding the meaning of statistical efficiency in a high-frequency sampling setting and analyzing the optimality of some of the commonly used statistical methods when applied to high-frequency data. The undertaken research responds to these two pressing problems. The project's outcomes have important applications in pricing of financial derivatives, calibration of financial models, monitoring of navigation system, intrusion detection in computer networks, and more. Educational impacts include providing summer research experiences for undergraduates and developing teaching/computational resources for interdisciplinary topics in statistics, probability, and mathematical finance. These activities involve graduate students and target the participation of underrepresented groups in sciences.