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.
