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

这个项目由四个主要主题组成，旨在创造跨越数学金融、数学流行病学和机器学习的综合知识。主题1探索了一种解决优化中时间不一致的新方法。例如，一个社会的长期财务规划必须面对时间不一致的问题(因为不同的世代可能不会就最佳财务规划战略达成一致)。将要开发的定点方法将为政策制定者提供一个方便的技术工具，以找到几代人之间的平衡，或为所有几代人所接受的战略。主题2将经济分析整合到传统的流行病建模中。它将捕捉到一种流行病如何改变个人的行为，以及这种行为的变化最终如何影响流行病的演变。其目的是促进政策制定，以预测人们对流行病的反应。主题3从两个互补的角度来探讨学生贷款：学生在多年的学习中如何积累债务，以及如何以具有成本效益的方式偿还债务。这项研究旨在为个人借款人提供真正的储蓄，为政策制定者提供具体的量化工具。主题4设计了新类型的梯度流来加强机器学习中的技术。它将为算法的设计提供严格的数学基础，并提供更大的灵活性来适应未知的动态。本科生和研究生将参与这个项目。该项目将通过将微分同胚随机流的理论与随机过程的收敛理论相结合来发展不动点方法(主题1)。这种联系将给出受控扩散泛函的新的收敛结果，并允许将平衡控制刻画为算子的不动点，并通过不动点迭代方便地找到。流行病的行为模型(主题2)依赖于易感人群、感染者和恢复者的消费行为的三人口模型。相关的哈密顿-雅可比-贝尔曼(HJB)方程由于可控跳跃而涉及到不寻常的非线性，这将通过粘性解技术的组合来解决。学生的债务积累和最优还款(主题3)将通过哈密顿量可能不允许最大化的平均场博弈和具有随机演化约束的随机水平控制问题来研究。解决这些问题将需要一种基于平均场博弈系统的广义解和随机水平随机庞特里亚金极大值原理的消失粘性方法。新类型的梯度流(主题4)将由(I)朗之万型McKean-Vlasov随机微分方程(SDE)或(Ii)朗之万型随机微分方程和受控扩散的耦合系统驱动。我们将研究SDE、非线性Fokker-Planck方程和HJB方程之间的相互联系，以揭示梯度流的不变分布。这将促进新的随机梯度下降方法在概率测量空间中进行静态和动态优化。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, 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.