Questions in Stochastic Process Theory Arising from Mathematical Finance

在他与让·雅各德合著的新书《过程的离散化》(Discretify of Process，2012)中，P.I.提议使用现代技术开发并解释统计检验，以确定随机波动率模型是否优于局部波动率模型，以及哪种风险资产的情况可能是这样。虽然有很多间接证据证明了这一点，但私家侦探还是建议系统地研究这个问题。此外，P.I.建议使用最近发展起来的过滤扩展理论中的技术来研究与内幕交易数学模型有关的问题。希望这种分析能够对试图确保公平金融市场的监管机构有所裨益，因为它显示了内幕交易如何影响内部人的风险中性衡量标准的计算，并使其不同于传统知情市场的风险中性衡量标准(从而影响期权价格)。股票价格演变的数学模型在“华尔街”上被广泛使用。虽然这些模型从经济上讲是合理的，但它们的种类很多，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，目前还没有任何综合形式的数学/统计技术来检查某一类模型是否比另一类模型更好。这笔赠款的目的是系统地制定这样的程序。这不仅应该为金融行业的从业者带来更准确的建模，而且应该有利于政府监管机构(如SEC、CFTC和美联储)努力将过度行为和腐败行为降至最低。这项研究的第二个目标是为内幕交易活动提供一个可行的数学模型。原则上，这应该会导致能够在内幕交易活动发生时实时发现它。

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.