Monte Carlo methods for fully non-linear PDEs and application to quantitative finance

蒙特卡罗方法为定量金融带来了一个新时代，为复杂模型中的风险定价提供了准确、可实现的数值方法。在一些模型中，对于某些类型的风险，偏微分方程（PDE）的解提供了风险的价格。定价偏微分方程的复杂性随着问题的维度和模型非线性成分的复杂性而增加。蒙特卡罗方法的一个主要特点是对问题的维度不像其他评价方法那样敏感，因为维度随着不相关风险因素的数量而增加。对于一类重要的偏微分方程，即完全非线性抛物方程，设计一个合适的蒙特卡罗方法是具有挑战性的。最近，PI和他的同事们的新工作打开了进步的大门。该奖项将利用蒙特卡罗方案在完全非线性抛物型偏微分方程的理论和实践方面的这些进展，并将在更简单的线性和半线性情况下纳入现有的想法。该项目将放宽以前必须做出的假设，并将产生更一般情况的方案，例如完全非线性抛物方程，既非凹也非凸的非线性项，或非lipschitzian项。本文将分析这些格式的收敛性，并给出收敛速度的渐近结果。此外，将考虑更有问题的偏微分方程，即简并偏微分方程。该奖项还将支持所谓的非单调蒙特卡罗方案的研究。这些方案在实践中被观察到比单调方案收敛得更快，但对这种改进的收敛性的严格数学验证在理论层面上仍然存在困难，这反过来又阻碍了进一步的广泛进展。理论发现将通过在实际数值格式中实施来验证。这一实施将涉及到夏季REU项目背景下的本科生。学生将学习具有广泛实际应用的最先进的数值技术。该奖项将支持在定量金融中有用的计算方法的发展。这项任务的传统方法被认为是相当准确的，但仅限于低复杂性的情况。众所周知，采用模拟随机实验的所谓蒙特卡罗方法可以避免这种瓶颈（在某些众所周知的准确性损失下），但到目前为止，它仅限于具有特殊（线性或近线性）结构的情况。这个项目将计算蒙特卡罗方案扩展到更一般的非线性结构的情况。该项目的结果将有助于数学家和金融工程师通过将其结果与市场模式进行比较来测试新模型，从而有助于扩展我们对金融市场的了解。具体来说，该项目将提供在模型中执行计算的有效方法。在直接应用层面，该项目将有助于在风险定价的实际方面取得进展，即对金融衍生品和风险相关产品进行更快、更可靠的评估。此外，在其他工程领域也有潜在的应用，例如图像处理中的降噪。这个研究项目的应用范围也有望吸引来自数学和其他学科的学生。该奖项将支持本科生通过暑期研究经验，并在一个令人兴奋和可访问的领域进行培训，并且从该研究中获得的结果将整合到课堂中。该奖项还将用于在数学研究界内外传播研究成果。

Monte Carlo methods have brought a new era to quantitative finance, resulting in accurate and implementable numerical methods for pricing risks in complicated models. In several models and for certain types of risks, the solution of a partial differential equation (PDE for short) provides the price of the risk. The complexity of the pricing PDE increases with the dimension of the problem and with the complication of the non-linear components of the model. One of the major features of Monte Carlo methods is to be less sensitive to the dimension of the problem than other evaluation methods, as the dimension grows with the number of uncorrelated risk factors. For a significant class of PDEs, i.e. for fully non-linear parabolic equations, designing an appropriate Monte Carlo method is challenging. Recently, new work by the PI and his colleagues opened the door to progress. This award will exploit these advances in theoretical and practical aspects of Monte Carlo schemes for fully non-linear parabolic PDEs and will also incorporate existing ideas in the simpler linear and semi-linear cases. The project will relax the assumptions that had to made previously and will result in schemes for more general situations, such as fully non-linear parabolic equations, nonlinear terms that are neither concave and nor convex, or non-Lipschitzian terms. The convergence of such schemes will be analyzed and asymptotic results for the rate of convergence will be derived. In addition, more problematic PDE's, i.e. degenerate PDEs will be considered. The award will also support the study of so called non-monotone Monte Carlo schemes. Such schemes are observed to converge faster in practice than monotone schemes, but the rigorous mathematical verification of this improved convergence still presents difficulties at the theoretical level, which in turn impedes further broad progress. The theoretical discoveries will be verified by implementing them in practical numerical schemes. This implementation will involve undergraduate students in the context of a summer REU program. Students will be trained in a class of state-of-the-art numerical techniques that have wide practical applications.The award will support the development of computational methods that are useful in quantitative finance. Traditional methods for this task are known to be quite accurate, but are limited to situations of low complexity. So-called Monte Carlo methods, which employ simulated random experiments, are known to avoid this bottleneck (at some well-known loss of accuracy), but so far have been limited to cases with special (linear or nearly linear) structure. This project will extend computational Monte Carlo schemes to situations with more general non-linear structure. The results of this project will help mathematicians and financial engineers to test new models by comparing their outcome with the patterns in the market and thus help to extend our knowledge about financial markets. Specifically, the project will provide efficient methods for performing computations in the models. At the immediate application level, the project will contribute to progress in the practical aspects of risk pricing, i.e. to develop faster and more reliable evaluations of financial derivatives and of risk related products. Moreover, there are potential applications also in other areas of engineering, for e.g. noise reduction in image processing. The breadth of applications of this research project is also expected to attract students from mathematics and other disciplines. The award will support undergraduate students through summer research experiences and train them in an exciting and accessible area, and the results obtained from this research will be integrated into classes. The award will also be used to disseminate the results within the mathematical research community and beyond.