Incomplete Markets and Financial Bubbles in Mathematical Finance

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The subject of speculative pricing in financial markets, leading to what is commonly known as bubbles, is a topic of current concern. Its importance is underscored by the huge housing market bubble, which crashed in 2008, and which thereby caused extensive financial suffering. This proposal aims to continue a study of the mathematical modeling of bubbles in financial markets. The mathematical basis for modeling speculative pricing is that it provides the opportunities to quantify when bubble pricing is occurring and how big (in an appropriate sense) the bubble is, and perhaps even more importantly, to identify when a bubble is occurring, or not. Early steps in this direction were burdened by rather severe restrictions in the generality of the mathematical models. In this research, the PI will continue the analysis in a more general setting. The key is to drop the standard restriction of what is known as a "complete market" in favor of the more realistic situation involving "incomplete markets."In the study of bubbles in incomplete markets, the role of strict local martingales will continue to be of paramount importance. The PI plans to tackle the issue of identifying models that lead to strict local martingales within incomplete markets, thus abandoning the wonderful but too simple framework first begun with the work of Delbaen and Shirakawa. The plan is to begin with the models of M. Musiela and P.L Lions, but then progress to multidimensional strict local martingales, using the theory of Lyapunov exponents as developed in the work of Narita, Khasminskii, Stroock, and Varadhan. The PI will also tackle some thorny numerical analysis issues created by the lack of linear growth, a problem inherent in the framework of strict local martingale models.