基于黎曼几何的无套利分析及其应用研究

In this project, we use Riemannian geometry theory, combining calculus of variations, fiber bundle theory, stochastic partial differential equation to study the problems of no-equilibrium geometric no-arbitrages and portfolio selections in incomplete financial markets. This project focuses on the characterizations of the no-equilibrium no-arbitrage and pricing mechanism of the frictional financial markets. We construct a gauge geometric frame of incomplete financial markets, design appropriate non-symmetric connections, and find the gauge invariants; The project studies mainly the effective frontier of portfolios and asset pricing models under non-monotone utility functions. We establish the M-V criterion under non-monotone utility functions, and derive a no-equilibrium capital asset pricing model; We study in depth the "Equity Premium" Puzzle. A no-equilibrium gauge geometric framework of cash flows + portfolio is constructed and the probability estimation of the minimizing variation between curvature and "equity premium" is obtained, we wish to crack further "equity premium" puzzle. These researches are expected to further clarify the intrinsic characteristics of incomplete frictional financial markets, and establish a no equilibrium geometric no-arbitrage framework to further develop the modern financial theory of incomplete markets. Because the investment portfolio and no arbitrage principle have important applications in, such as derivatives pricing, financial risk management, etc., applied mathematics, finance, securities and other fields, and which has provoked the extensive interests in the past two decades, therefore, it is indispensable to carry out the research in this field.

本课题拟用黎曼几何理论，结合变分法、纤维丛理论、随机微分方程等理论，研究不完全摩擦金融市场之非均衡几何无套利原理与投资组合问题。重点研究摩擦金融市场的非均衡无套利刻画及其定价机制。构建不完全金融市场的规范几何结构，设计合适的非对称联络,给出规范不变量；主要研究非单调效用函数下的投资组合有效前沿与资产定价问题。建立非单调效用下的M-V准则，给出非均衡资本资产定价模型；深入研究“股权溢价”问题。建立现金流+投资组合的非均衡规范几何框架，给出曲率与“股权溢价”最小偏差概率估计，破解“股权溢价”之谜。这些研究可望进一步弄清不完全摩擦金融市场的内蕴特征，建立非均衡几何无套利分析框架，进一步发展不完全市场之现代金融理论。由于无套利原理与投资组合在衍生品定价、金融风险管理等诸多应用数学、金融学、证券等领域中都有重要的应用，在过去的二十余年来引起了广泛关注，所以开展这一领域的研究非常必要。

项目组利用黎曼几何理论，结合变分法、纤维丛理论、随机分析等理论，研究不完全摩擦金融市场之非均衡几何无套利原理与资产定价问题。探究并发现了单参数变换群与折现因子之间的一一对应关系；借助李导数建立了一类基于常比例投资策略调整的摩擦金融市场中几何无套利原理；借助半对称联络变换群理论，建立了非常比例投资策略调整的摩擦金融市场之几何无套利原理；进而根据此几何无套利原理，给出了资产定价基本定理。该研究解决了非单调效用函数下的投资组合有效前沿存在性问题与非均衡资产定价模型。这一结果也部分回答了股权溢价问题。. 本项目建立并完善了摩擦金融市场之几何无套利原理以及基于几何无套利原理的资产定价模型。已发表或录用论文40余篇，协助数学与统计学院组织了一次《几何PDE》国际学术会议、参加几何分析与金融数学国际研讨会等会议6场。待出版专著《几何无套利分析》一本。

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