Questions in Probability Relating to Mathematical Finance

该奖项将支持PI对金融泡沫和股票市场内幕交易数学模型的研究。使用数学模型，人们可以获得对市场的直觉知识通常无法获得的洞察力，这大致类似于人们如何通过望远镜而不是肉眼观察天空来看到更多细节并学习新事物。PI将处理信用风险模型，以及从业人员（例如银行和投资公司）目前使用的方法来计算所涉及的风险。该研究计划提出了一个明智的推测，即当泡沫存在时，美国和世界各地的金融建模者使用的标准近似值实际上比目前认为的要糟糕得多。这项研究将利用概率论中与不确定性下信息的可用性和流动有关的微妙技术;因此，将建立新的数学结果，加深我们对概率的理解。对金融业的更广泛影响可能会使银行家、投资者和监管机构能够利用工具来了解并因此提高国家和全球金融稳定性。 研究金融风险数学模型的经典方法是使用简化形式的模型，并试图近似风险率，这给出了在给定时间即将发生违约的可能性。PI旨在研究在存在金融泡沫的情况下，这些简化形式模型的数值近似方法收敛速度有多快。这涉及到随机微分方程的解的数值分析时，系数既不是Lipschitz连续的，也没有线性增长的空间变量，创造技术挑战，应该有显着的数值影响。

The award will support the PI's research on mathematical models of financial bubbles and of insider trading in the stock market. Using mathematical models one can obtain insights not normally available to mere intuitive knowledge of the markets, in rough analogy to how one can see more details and learn new things by looking at the heavens through a telescope rather than with the naked eye. The PI will treat models of credit risk, and methods currently in use by practitioners (for example banks and investment houses) to calculate the risk involved. The research plan makes the informed conjecture that when bubbles are present, the standard approximations used by financial modelers in the US and around the world are in fact significantly worse than is currently believed. The research will draw on delicate techniques in probability theory which have to do with the availability and flow of information under uncertainty; as a consequence, new mathematical results will be established which deepen our understanding of probability. The broader impact which will apply to finance will potentially result in tools available to bankers, investors, and regulators to understand and therefore increase national and global financial stability. The classical way of looking at mathematical models of financial risk uses reduced-form models and attempts to approximate the hazard rate, which gives the likelihood of imminent default at a given time. The PI intends to investigate how fast numerical approximation methods converge for these reduced-form models in the presence of financial bubbles. This involves the numerical analysis of solutions of stochastic differential equations when the coefficients are neither Lipschitz-continuous, nor have linear growth in the space variable, creating technical challenges which should have significant numerical implications.