Stochastic Process Research Inspired by Problems from Mathematical Finance

首席调查员普罗特使用了这样的想法，即sigma-代数的过滤可以对可用信息的历史进行建模。在应用中，拥有不同级别的可用信息有时是重要的，这反映在这些西格玛代数(或可观测事件的历史)之间的包容关系中。例如，在经济学中，有一个数学条件表明不存在套利机会(即在不承担任何风险的情况下获利的可能性)，而这个条件是否成立，取决于西格玛代数的精细结构。这一现象被证明是存在的，并将得到系统的调查。在提案的第二部分，P.I.建议研究随机过程各个方面的统计估计的离散化程序。这些技术已经导致了一些微妙的结果，例如统计测试，以查看动态演变的数据是否以连续流的形式到达，或者是否具有内在的跳跃。这将是一项大规模的研究，最终将出版一本关于这个主题的书。

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.