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

本项目研究随机动力系统的随机分析和数值算法，以及它们在机器学习和金融中的应用。第一部分集中在机器学习/数据科学的基础上，它保证了数值算法(如随机梯度下降、马尔可夫链蒙特卡罗)在非凸优化和多峰分布抽样中的理论收敛。该项目将开发算法来解决大数据和工程中的此类问题，包括人工智能安全问题中的不确定性量化、控制机器人运动和图像处理。第二部分重点研究了数学金融中的随机模型和期权/资产定价中的算法设计。这一部分的应用针对的是具有粗略波动性的路径依赖期权定价的有效算法，预计这将对一些面向计算的金融工具产生重大影响，例如涉及粗略波动率和高频数据的基于模型的算法交易。该项目将为研究生和本科生提供支持和研究机会。这个项目中的随机系统具有简并、平均场或非马尔科夫性质。在第一部分中，PI将研究高度退化的平均场随机动力系统的亚矫顽力(即收敛于平衡)及其在机器学习中的算法设计中的应用。其中一个主题将集中于一般简并/平均场系统的(非)渐近分析及其指数收敛到平衡的速度(例如，Vlasov-Fokker-Planck方程；高阶幂零李群上的朗之万动力学)。作为这种动力学收敛的应用，PI将设计专注于机器学习中的非凸优化和分布采样的算法。在第二部分，PI将研究非马尔科夫随机动力系统，该系统捕捉金融市场的路径依赖和平均场特征。主题包括路径相关的偏微分方程组、随机Volterra积分方程组、条件平均场SDE和Volterra签名。PI侧重于解决基本问题，包括粗糙波动率模型和条件平均场SDE的密度以及Volterra签名的结构。此外，PI侧重于使用Volterra签名和深度神经网络来设计高效的数值算法。这些算法旨在解决路径依赖的偏微分方程、路径依赖的期权定价和最优停车/切换问题。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.