Questions in Stochastic Process Theory Arising from Mathematical Finance

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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.