Incomplete Markets and Financial Bubbles in Mathematical Finance

金融市场中的投机性定价问题，导致了通常所说的泡沫，是当前令人担忧的话题。2008年，巨大的房地产市场泡沫破裂，并由此造成了广泛的金融灾难，突显了它的重要性。这项提议旨在继续研究金融市场泡沫的数学建模。为投机性定价建模的数学基础是，它提供了量化泡沫定价何时发生以及泡沫有多大(在适当意义上)的机会，或许更重要的是，它提供了确定泡沫何时发生或没有发生的机会。在这个方向上的早期步骤受到了数学模型一般性方面相当严格的限制。在本研究中，PI将在更一般的背景下继续分析。关键是要放弃所谓“完全市场”的标准限制，取而代之的是涉及“不完全市场”的更现实的情况。在研究不完全市场中的泡沫时，严格局部鞅的作用仍将是至关重要的。PI计划解决在不完全市场中识别导致严格局部鞅的模型的问题，从而放弃最初由Delbaen和Shirakawa的工作开始的精彩但过于简单的框架。我们的计划是从M.Musiela和P.L Lions的模型开始，然后利用成田、Khasminskii、Stroock和Varadhan的工作中发展的Lyapunov指数理论，发展到多维严格局部鞅。PI还将解决由于缺乏线性增长而产生的一些棘手的数值分析问题，这是严格的局部鞅模型框架中固有的问题。

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.