Topics in Stochastic Control: Finance, Epidemics, and Machine Learning

随机控制主题：金融、流行病和机器学习

• 批准号: 2109002
• 负责人: Yu-Jui Huang
• 金额: $ 27.34万
• 依托单位: University of Colorado at Boulder
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-08-01 至 2024-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2109002&HistoricalAwards=false
• 关键词: Topics; Stochastic; Control; Finance; Epidemics

This project, consisting of four main topics, aims to create integrated knowledge across mathematical finance, mathematical epidemiology, and machine learning. Topic 1 explores a new method to resolving time inconsistency in optimization. For example, long-term financial planning in a society must confront time inconsistency (as different generations may not agree on an optimal financial planning strategy). The fixed-point approach to be developed will provide a convenient technical tool for policymakers to find equilibria between generations, or strategies acceptable to all generations. Topic 2 integrates economic analysis into traditional epidemic modeling. It will capture how an epidemic alters individuals' behaviors and how this change of behaviors ultimately influences the epidemic's evolution. The aim is to facilitate policymaking that anticipates people's reactions to an epidemic. Topic 3 approaches student loans from two complementary angles: how the debt accumulates over a student's years of study and how to repay the debt in a cost-efficient way. This study aims to provide individual borrowers with real savings and policymakers with concrete quantitative tools. Topic 4 devises new types of gradient flows to strengthen techniques in machine learning. It will provide rigorous mathematical foundations for the design of algorithms and more flexibility to accommodate unknown dynamics. Undergraduate and graduate students will be involved in this project. The project will develop the fixed-point approach (Topic 1) by merging theory for stochastic flows of diffeomorphisms with convergence theory for stochastic processes. Such a link will give new convergence results for functionals of controlled diffusions and would allow equilibrium controls to be characterized as fixed points of an operator and conveniently found via fixed-point iterations. Behavioral models for epidemics (Topic 2) rely on a three-population model of consumption behaviors of the susceptible, infected, and recovered. The associated Hamilton-Jacobi-Bellman (HJB) equation involves unusual nonlinearity due to controllable jumps, which will be approached by a combination of viscosity solutions techniques. A student's debt accumulation and optimal repayment (Topic 3) will be investigated through a mean field game whose Hamiltonian may not admit a maximizer and a random-horizon control problem with a stochastically evolving constraint. Resolving them will demand a vanishing viscosity method based on generalized solutions to a mean field game system and a random-horizon stochastic Pontryagin maximum principle. New types of gradient flows (Topic 4) will be driven by (i) a Langevin-type McKean-Vlasov stochastic differential equation (SDE) or (ii) a coupled system of a Langevin SDE and a controlled diffusion. Interconnections among SDEs, nonlinear Fokker-Planck equations, and HJB equations will be investigated to uncover the gradient flows' invariant distributions. This will facilitate new stochastic gradient descent approaches to both static and dynamic optimization in the space of probability measures.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.

该项目由四个主要主题组成，旨在在数学金融，数学流行病学和机器学习中创建综合知识。主题1探讨了一种解决优化时间不一致的新方法。例如，社会中的长期财务计划必须面对时间不一致（因为不同的一代可能不同意最佳的财务计划策略）。要开发的定点方法将为决策者提供方便的技术工具，以在世代之间找到平衡或所有世代可接受的策略。主题2将经济分析纳入传统的流行模型。它将捕捉流行病如何改变个人的行为以及这种行为的改变最终如何影响流行病的进化。目的是促进人们预期人们对流行病的反应的决策。主题3从两个互补角度汇款：学生贷款如何在学生的学习年中积累，以及如何以经济高效的方式偿还债务。这项研究旨在为个人借款人提供真正的储蓄和决策者，并提供具体的定量工具。主题4设计新型的梯度流以增强机器学习的技术。它将为设计算法的设计提供严格的数学基础，并具有更大的灵活性以适应未知动态。本科生和研究生将参与该项目。该项目将通过将理论与随机过程的融合理论合并为差异理论来开发定点方法（主题1）。这样的链接将为受控扩散功能提供新的收敛结果，并允许将平衡控件描述为操作员的固定点，并通过固定点迭代方便地找到。流行病的行为模型（主题2）依赖于易感，感染和恢复的三个人口模型。相关的Hamilton-Jacobi-Bellman（HJB）方程涉及由于可控跳跃而引起的异常非线性，这将通过粘度解决方案技术的组合来接近。学生的债务积累和最佳还款（主题3）将通过平均野战游戏进行调查，该野战游戏可能不会承认最大化器和随机发展的随机控制问题，并具有随机发展的约束。解决它们将要求基于平均野外游戏系统的通用解决方案和随机匹配的随机pontryagin pontryagin最大原理，要求消失的粘度方法。新型的梯度流（主题4）将由（i）langevin型McKean-Vlasov随机微分方程（SDE）或（ii）langevin SDE和受控扩散的耦合系统驱动。将研究SDE，非线性Fokker-Planck方程和HJB方程之间的互连，以发现梯度流的不变分布。这将促进概率措施空间中静态和动态优化的新的随机梯度下降方法。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和更广泛影响的评估评估标准来通过评估来支持的。

项目成果

GANs as Gradient Flows that Converge

• DOI: 10.48550/arxiv.2205.02910
• 发表时间: 2022-05
• 期刊: ArXiv
• 作者: Yu‐Jui Huang;Yuchong Zhang

Minimizing the Repayment Cost of Federal Student Loans

最大限度地降低联邦学生贷款的偿还成本

• DOI: 10.1137/22m1505840
• 发表时间: 2022
• 期刊: SIAM Review
• 影响因子: 10.2
• 作者: Guasoni, Paolo;Huang, Yu-Jui
• 通讯作者: Huang, Yu-Jui

On characterizing optimal Wasserstein GAN solutions for non-Gaussian data

• DOI: 10.1109/isit54713.2023.10206785
• 发表时间: 2023-06
• 期刊: 2023 IEEE International Symposium on Information Theory (ISIT)
• 作者: Yujia Huang;Shih-Chun Lin;Yu-Chih Huang;Kuan-Hui Lyu;Hsin-Hua Shen;Wan-Yi Lin