全非线性Feynman-Kac公式及其应用的若干研究

Being an important and non-trivial generalization of the classical probability theory, the recently developed nonlinear expectation theory provides a powerful tool for the study of financial asset pricing and risk measure under Knightian uncertainty. Moreover, it is significant that this theory allows for Feynman-Kac type representation for nonlinear HJB equations rather than linear ones, which leads to a brand-new probabilistic approach to fully nonlinear PDEs. .Based on the fully nonlinear Feynman-Kac formula, our research will be developed in the following aspects. First, we are going to consider the well-posedness of the so-called fully nonlinear parabolic path-dependent partial differential equations (PDEs) through stochastic analysis and PDE techniques. In particular, our plan is to establish the corresponding fully nonlinear Feynman-Kac formula in non-Markovian case, which would be a potential answer to one of Prof. Peng’s open questions posed in the International Congress of Mathematicians 2010 (ICM 2010). Then we will investigate the limit theorems for stochastic processes in the nonlinear expectation framework such as averaging principle. This research is the fundamental for the study of asymptomatic properties of solutions to fully nonlinear parabolic PDEs, for example, the homogenization. Finally, we shall study the pricing theory for (path-dependent) derivatives under models uncertainty. Furthermore, numerical algorithm of path-dependent PDEs will also be discussed for solving real financial problems in order to improve the methods of mathematical finance..In conclusion, this project is motivated by the open problem in the connection between the nonlinear expectation and PDEs theory. By conducting high-level research, our aim is to open new perspectives for nonlinear stochastic analysis and to reinforce the framework of nonlinear expectation including both theory and practical application.

非线性(数学)期望理论不仅很好地刻画了模型不确定性下的资产定价与风险度量等经济金融问题，还非平凡地推广了经典概率论。特别地，它将经典的线性Feynman-Kac公式推广至全非线性情形，提供了研究全非线性抛物型偏微分方程的概率方法。为此，本项目将基于全非线性Feynman-Kac公式这一纽带，通过随机分析方法以及偏微分方程技术的综合应用，研究全非线性抛物型轨道偏微分方程的适定性，以建立非马尔可夫情形下的全非线性Feynman-Kac公式，解决彭实戈院士在2010年数学家大会上提出的这一公开问题；探索非线性期望框架下的随机过程极限理论，以分析全非线性抛物型偏微分方程的均匀化等渐近极限问题，丰富和完善偏微分方程理论；讨论模型不确定性下的衍生产品定价以及数值计算等实际金融问题。我们希望，通过该项目的研究得到一些随机分析以及金融数学中具有国内外领先水平的原创性成果，促进非线性期望理论的发展与应用。

非线性数学期望理论提供了研究全非线性Feynman-Kac公式的新思路，刻画了模型不确定性下的资产定价与风险度量等经济金融问题。为此本项目深入地研究了以G-正倒向随机微分方程为核心的全非线性Feynman-Kac公式、全非线性轨道偏微分方程、全非线性抛物型偏微分方程的渐近化及相关问题。我们通过随机分析方法以及偏微分方程技术的综合应用，建立了一般HJBI方程的全非线性Feynman-Kac公式，并讨论了HJBI方程的粘性解的存在唯一性理论及其在模型不确定性下金融市场中资本资产的风险度量问题中的应用；创立了G-期望框架下的遍历倒向随机微分方程理论，发展了G-随机微分方程的奇异摄动理论，解决了全非线性抛物型偏微分方程的均匀化等渐近极限问题，突破了经典概率论中线性这一框架限制。除此之外，我们还研究了非线性框架下布朗运动的容度估计问题，随机微分方程的生存性问题，均值反射倒向随机微分方程理论及其在风险管理约束下衍生产品的定价问题中的应用，得到了一系列随机分析、随机控制和金融数学的国际前沿、国内领先的应用基础成果。

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