Questions in Probability Relating to Mathematical Finance

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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.