Path-Dependent Partial Differential Equations and Optimal Control

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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.