Hypoelliptic and Non-Markovian stochastic dynamical systems in machine learning and mathematical finance: from theory to application

机器学习和数学金融中的亚椭圆和非马尔可夫随机动力系统：从理论到应用

• 批准号: 2306769
• 负责人: Qi Feng
• 金额: $ 16.52万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-06-01 至 2024-03-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2306769&HistoricalAwards=false
• 关键词: Hypoelliptic; Non; Markovian; stochastic; dynamical

This project investigates stochastic analysis and numerical algorithms for stochastic dynamical systems, together with their applications in machine learning and finance. The first part focuses on the foundations of machine learning/data science, which guarantees the theoretical convergence of numerical algorithms (e.g., stochastic gradient descent, Markov Chain Monte Carlo) in non-convex optimization and multi-modal distribution sampling. This project will develop algorithms to solve such problems in big data and engineering, which include uncertainty quantification in AI safety problems, control robotics motions, and image processing. The second part focuses on the stochastic models in mathematical finance and algorithm designs in option/asset pricing. The applications in this part target efficient algorithms for path-dependent option pricing with rough volatilities, which are expected to significantly impact some computation-oriented financial instruments, such as model-based algorithm trading involving rough volatility and high-frequency data. This project will provide support and research opportunities for graduate and undergraduate students. The stochastic systems in this project possess degenerate, mean-field, or non-Markovian properties. In the first part, the PI will study the "hypocoercivity" (i.e., convergence to equilibrium) for highly degenerate and mean-field stochastic dynamical systems and their applications to algorithms design in machine learning. One of the proposed topics will focus on the (non)-asymptotic analysis of the general degenerate/mean-field system and its exponential convergence rate to the equilibrium (e.g., Vlasov-Fokker-Planck equations; Langevin dynamics on higher order nilpotent Lie groups). As applications of the convergence of such dynamics, the PI will design algorithms focusing on non-convex optimizations and distribution samplings in machine learning. In the second part, the PI will study non-Markovian stochastic dynamical systems capturing path-dependent and mean-field features of the financial market. The topics include path-dependent PDEs, stochastic Volterra integral equations, conditional mean-field SDEs, and the Volterra signatures. The PI focuses on addressing the fundamental issues, including the density for the rough volatility model and conditional mean-field SDEs and the structure of Volterra signatures. Furthermore, the PI focuses on designing efficient numerical algorithms using the Volterra signature and deep neural networks. These algorithms target solving path-dependent PDEs, path-dependent option pricing, and optimal stopping/switching problems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目研究了随机动力学系统的随机分析和数值算法，以及它们在机器学习和融资中的应用。第一部分的重点是机器学习/数据科学的基础，该基础保证了非convex优化和多模式分布采样的数值算法（例如随机梯度下降，马尔可夫链蒙特卡洛）的理论收敛。该项目将开发算法来解决大数据和工程中的此类问题，其中包括AI安全问题，控制机器人动作和图像处理中的不确定性量化。第二部分侧重于选项/资产定价中数学金融和算法设计中的随机模型。该部分的应用程序有效算法在路径依赖性期权定价中具有粗大波动性，预计将显着影响某些面向计算的金融工具，例如基于模型的算法交易，涉及粗糙的波动性和高频数据。该项目将为研究生和本科生提供支持和研究机会。该项目中的随机系统具有退化，平均场或非马克维亚特性。在第一部分中，PI将研究高度退化和平均场随机动力学系统及其在机器学习中的算法设计中的“低调”（即汇合到平衡）。拟议的主题之一将集中于对一般退化/平均场系统的（非） - 反应分析及其指数收敛速率（例如Vlasov-Fokker-Planck方程； vlasov-fokker-Planck方程； langevin动力学上的高阶Nilpotent Lie lie组）。作为这种动力学收敛的应用，PI将设计针对机器学习中非凸优化和分配采样的算法。在第二部分中，PI将研究非马克维亚随机动力学系统，以捕获金融市场的路径依赖性和均值特征。主题包括依赖路径的PDE，随机Volterra积分方程，有条件的平均场SDE和Volterra特征。 PI致力于解决基本问题，包括粗糙波动率模型的密度和有条件的平均场SDE和Volterra特征的结构。此外，PI专注于使用Volterra特征和深神经网络设计有效的数值算法。这些算法针对求解路径依赖的PDE，依赖路径的期权定价和最佳停止/开关问题的算法。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛的影响来通过评估来获得支持的。