Stochastic Process Research Inspired by Problems from Mathematical Finance

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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.