The objective of this proposal is to develop a range of specification tools for continuous-time models sampled at high frequency. Continuous-time stochastic processes known as semimartingales are effectively, up to technical conditions, the largest class of models that can be used to represent asset prices while still precluding arbitrage opportunities. As a result, they form the foundation of most theoretical asset pricing models. While these models have been widely studied in the probability literature, relatively few econometric and statistical procedures exist to determine the relative importance of different components of the model, test for their very presence, or look into some of their finer characteristics, such as the degree of activity of jumps.
Semimartingales have different components: a possibly stochastic drift, a continuous martingale part which is the integral of a possibly stochastic volatility process with respect to a Brownian motion, and jumps, with a possibly stochastic Lévy measure. Ultimately, the objective of this proposal is to develop a nearly complete menu of specification tools for discretely-sampled semimartingales, and apply them on high frequency data: _ Which component(s) of the model should be included, given that some other components are already present; _ Which component(s) of the model can be distinguished from which existing components, and under what restrictions on the nature of the component(s); _ What are some of the finer characteristics of the jump components, such as the degree of activity of jumps, whether they have finite or infinite activity, etc.
Broader Impacts: While many of the examples and motivation for the work that are described in this proposal come from economics and finance, this proposal is fundamentally about developing methodologies that are applicable to generic semimartingales. Semimartingales are some of the most common tools from probability used to model stochastic processes. One can hope that some of the econometric and statistical methods developed here can find applications in physics, engineering or seismology, which are all fields where observations are also available at very high frequencies.
The project integrates research and education by working closely with students and funding them in the form of summer research assistantships, creating datasets and developing publicly available computer code (both made available through the web, as has been the case for past projects) for each of the main research endeavors funded by this proposal. Various research findings and examples will be simplified and taught in both undergraduate and graduate courses. In particular, they will be used in supervising undergraduate senior theses and Ph.D. theses. Postdoctoral fellows and underrepresented groups will be trained as a part of the research covered by this proposal. The results will be disseminated broadly through presentations at seminars, conferences and professional association meetings.
Finally, this project involves collaborations with colleagues in different disciplines (economics, statistics, probability) and academic institutions in the U.S. (Per Mykland, a statistician at the University of Chicago, Joon Park, an economist at Texas A&M University, Lan Zhang, a statistician at UIC) and abroad (Jean Jacod, a probabilist at Université Paris VI).
