Stochastic Process Research Inspired by Problems from Mathematical Finance

首席研究员Protter使用的想法是，对西格玛代数的过滤可以模拟可用信息的历史。在应用中，有时具有不同级别的可用信息是很重要的，这反映在这些sigma代数（或可观察事件的历史）之间的包含关系中。例如，在经济学中，有一个数学条件表明不存在套利机会（在不承担任何风险的情况下获利的可能性），这个条件是否成立，取决于sigma代数的精细结构。这一现象已被证实会发生，并将被系统地研究。在建议的第二部分，P.I.建议研究离散程序的统计估计随机过程的各个方面。这些技术导致了一些微妙的结果，例如统计测试，以查看动态发展的数据是否以连续流的形式到达，或者具有内在的跳跃。这将是一项大规模的研究，并将出版一本有关该主题的书。

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.