Questions in Stochastic Process Theory Arising from Mathematical Finance

私家侦探建议开发统计测试，使用他与Jean Jacod合著的最新著作《离散化过程》（Springer，2012）中开发和解释的现代技术，以确定随机波动率模型是否上级局部波动率模型，以及哪种风险资产可能是真的。虽然有很多间接证据表明情况确实如此，但P.I.建议系统地研究这个问题。此外，PI。建议使用最近开发的技术在理论的扩张过滤研究问题的数学模型的内幕交易。人们希望，这样的分析可能是有益的监管机构试图确保公平的金融市场，通过显示内幕交易如何影响内部人员的风险中性措施的计算，并使其不同（从而影响期权价格）从传统的知情市场的风险中性措施。股票价格演变的数学模型在“华尔街”被广泛使用。虽然这些模型是通过经济推理来证明的，但它们的种类繁多，从业者试图使用他们认为符合现实的模型。这是一个困难的过程，并且用于检查一类模型是否优于另一类模型的数学/统计技术目前还不存在任何全面的形式。这笔赠款的目的是有系统地发展这种程序。这不仅会为金融业从业者带来更准确的建模，也会使政府监管机构（如SEC、CFTC和联邦）在尽量减少过度行为和腐败行为方面受益。本研究的第二个目标是提供一个可行的数学模型的内幕交易活动。原则上，这应该导致能够发现内幕交易活动，因为它发生在真实的时间。

The P.I. proposes to develop statistical tests, using modern techniques as developed and explained in the recent book Discretization of Processes, Springer, 2012 that he co-authored with Jean Jacod, to determine if stochastic volatility models are superior to local volatility models, and for which kind of risky assets that might be true. While there is much indirect evidence this is the case, the P.I. proposes systematically to examine the question. In addition, the P.I. proposes to use recently developed techniques in the theory of the expansion of filtrations to study questions concerning mathematical models of insider trading. It is hoped that such an analysis could be of benefit to regulators trying to ensure equitable financial markets, by showing how insider trading affects the calculation of the risk neutral measure of the insider, and renders it different (thereby affecting option prices) from the risk neutral measure of the traditionally informed market. Mathematical models of the evolution of stock prices are widely used on "Wall Street." While the models are justified by economic reasoning, there is a wide variety of them, and practitioners try to use models that they think correspond to reality. This is a difficult procedure, and mathematical/statistical techniques to check to see if one class of models is better than an alternative class currently do not exist in any comprehensive form. It is the purpose of this grant to develop systematically such procedures. This should lead to more accurate modeling not just for practitioners of the financial industry, but also it should benefit government regulators (such as the SEC, the CFTC, and the Federal Reserve) in their attempts to minimize excesses and corrupt practices. A second goal of this research is to provide a workable mathematical model of insider trading activity. In principle this should lead to the ability to detect insider trading activity as it happens in real time.