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

受该领域最新理论研究成果的启发，研究者确定了Levy过程在短时间内渐近行为的一些关键开放问题，并将其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。 对于有限的随机样本，方法，如最大似然估计和累积和（累积和）顺序规则是已知的是最佳的处理上述两个问题。虽然人们期望当Levy过程的连续观测之间的时间跨度缩小到零时，最优性将被保持，但存在重要的例子表明情况并非总是如此。这些反直觉结果背后的奥秘与Levy过程在短时间内的“精细”分布性质密切相关。而不是直接攻击两个连续时间提出的问题，研究人员建立在离散时间的研究类似的问题，并填写在无限的时间连续分析其演变时，连续观测之间的时间跨度越来越小。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的统计方法的性能，而不是连续时间观测，这可以说是永远无法获得的。对Levy过程的关注是出于这样一个事实，即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。扩展到其他多因素随机模型驱动的利维过程也被考虑。自然和社会现象的自动高频监测系统越来越多地用于工程应用、金融市场和环境研究。因此，对于这些系统所产生的高频数据，越来越需要有效和准确的统计和计算方法。两个重要的问题出现了这种需要：理解的意义，统计效率在高频采样设置和分析的最优性时，一些常用的统计方法应用于高频数据。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，并为统计，概率和数学金融的跨学科主题开发教学/计算资源。这些活动有研究生参与，目标是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.