The Principal Investigator Protter uses the idea that a filtration of sigma-algebras can model a history of available information. In applications, it is sometimes of importance to have different levels of available information, and this is reflected in containment relations among these sigma algebras (or histories of observable events). In economics, for example, there is a mathematical condition which indicates an absence of arbitrage opportunities (the possibility to make a profit without taking any risk), and this condition can hold or not, depending on the fine structure of the sigma algebras. This phenomenon is shown to happen, and will be investigated systematically. In a second part of the proposal, the P.I. proposes to study discretization procedures for the statistical estimation of various aspects of stochastic processes. These techniques have led to some delicate results, such as statistical tests to see whether or not dynamically evolving data arrives in a continuous stream, or has intrinsic jumps. This will be a massive study, resulting in the publication of a book on the subject.
The results from the NSF Grant "Stochastic Process Research Inspired by Problems from Mathematical Finance" include a detailed study of how to expand models of observable events as time evolves, while nevertheless maintaining key mathematical properties. The idea of information permeates applications, and one particularly powerful example is that of modeling 'inside information," such as the infamous examples of "insider trading" in the financial world. It is hoped these results will eventually lead to more effective ways to regulate the financal markets. In another mathematical result with implications for the financial markets, we have performed a detailed mathematical model of financial bubbles, and proposed a method to detect their existence in real time. An important technique developed during this grant is to analyze tick data from the financial markets to esitmate the diffusion coefficient in the mathematical model of a stock price. The figure illustrates how we were able to do this, using several different procedures, each represented by its own color: In terms of an intellectual impact, we have trained 5 PhD students during the course of this award, and begun training of three more. We have taught short courses and given lectures widely across the world, including lectures open to the general public, from Louisville, Kentucky to Nice, France. In addition, we have organized professional conferences in New York, Berkeley, and Singapore.
