Stochastic Optimal Control with High Dimensional Data

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Optimal control has been a central tool in many groundbreaking technological advances starting with the moon-landing problem to recent developments in machine learning. Impressive advances in the training of neural networks now allow for further exciting complex and realistic applications. This project aims to leverage these modern optimization techniques to build an almost data-driven theory and to reduce the potentially catastrophic model risk that was prevalent in the recent financial crisis. Additionally, as machine learning methodology is becoming the dominant paradigm in many industries, education on these topics is vital to the economy of the nation. Towards this goal, students from all levels will be integrated into this research providing them with well-rounded training on these omnipresent computational practices. Technically, several classes of problems will be investigated in-depth to highlight and resolve different difficulties that the general theory faces. The main study will be a general high-level analysis of recent numerical experiments based on the efficient training of deep neural networks. Optimal control of McKean-Vlasov jump-diffusions will be analyzed to provide a concrete setting. These problems are naturally set in the infinite-dimensional Wasserstein-type spaces and pose many interesting questions, including the construction of high-dimensional but tractable approximations. Similar questions also arise in optimal transport problems related to optimization with model uncertainty and risk management problems in quantitative finance with many underlying risk factors, and they have broad applicability.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.