Path-Dependent Partial Differential Equations and Optimal Control

这项研究将推进最优控制理论，以了解实际和复杂的应用场景。这项研究将有可能提供更好甚至最佳的决策程序，特别是与影响许多利益攸关方的问题有关，如养老基金投资和金融风险管理。该项目将为研究生提供培训机会，并为STEM学生提供在金融行业建立职业生涯的机会。该项目包括三个部分。在第一部分中，研究者将研究涉及非马尔可夫分段确定过程和相关的非局部Hamilton-Jacobi-Bellman方程的最优控制问题。应用是数学金融中的最优执行或清算问题，具有更丰富的模型类别。例如，可以并入非马尔可夫霍克斯过程。这些程序更准确地描述了相关的财务数据。第二部分研究了哈密顿量在梯度上可以二次甚至超二次增长的路径依赖型Hamilton-Jacobi方程。这些方程对于具有无界控制的最优控制问题是很重要的。研究人员将开发新的非光滑解决方案的概念，并建立关于这些概念的适定性结果。在本计画的第三部分，研究者将完整地进行局部单调发展方程最优控制的路径相依动态规划方法。这一大类方程包括二维Navier-Stokes方程和驯服的三维Navier-Stokes方程。此外，这项研究为无限维控制问题的最优综合这一大规模开放问题提供了一种新的方法。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

This research will advance optimal control theory to understand realistic and complex scenarios in applications. The research will potentially provide better or even optimal decision-making procedures, especially relevant for problems that affect many stakeholders such as pension fund investments and financial risk management. The project will provide training opportunities for graduate students and opportunities for STEM students to build their careers in the financial industry. The project consists of three parts. In the first part, the investigator will study optimal control problems involving non-Markovian piecewise deterministic processes and the associated non-local Hamilton-Jacobi-Bellman equations. Applications are optimal execution or liquidation problems in mathematical finance with richer classes of models. For example, non-Markovian Hawkes processes can be incorporated. Those processes provide a more accurate description of the relevant financial data. The second part deals with path-dependent Hamilton-Jacobi equations whose Hamiltonians can have quadratic or even super-quadratic growth in the gradient. Those equations are important for optimal control problems with unbounded controls. The investigator will develop new notions of non-smooth solutions and establish well-posedness results with respect to those notions. In the third part of this project, the investigator will completely carry out a path-dependent dynamic programming approach for the optimal control of locally monotone evolution equations. This large class of equations covers the two-dimensional Navier-Stokes equations and the tamed three-dimensional Navier-Stokes equations. In addition, this research provides a new line of methodologies for the largely open problem of optimal synthesis for infinite-dimensional control 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.