Applications of infinite dimensional PDEs to transport, finance, and games

This interdisciplinary research project aims to develop new mathematical tools based on partial differential equations (PDEs) and apply them to address practical challenges in mathematical finance and game theory. PDEs serve as fundamental mathematical models for understanding complex phenomena in these fields, ranging from optimization strategies in uncertain markets to analyzing strategic interactions in competitive environments. The unique aspect of this research lies in its focus on PDEs formulated in infinite-dimensional spaces, which present both theoretical challenges and opportunities for developing innovative solutions tailored to real-world financial and strategic decision-making scenarios. This award will also provide opportunities for students to be involved in the research projects. In the first part of the project, infinite-dimensionality arises because one of the state variables is in the Wasserstein space of probability measures. Some of the equations to be studied are motivated by the large population limit of the equilibrium among N weakly interacting symmetric agents. The primary goal is to use these partial differential equations to demonstrate sharp convergence rates to the large population limit. A second class of equations in the Wasserstein space is inspired by causal optimal transport problems and the control of the solution of Kushner's equation for optimal filtering. While these transport problems were initially introduced to measure distances between the laws of stochastic processes, the project aims to establish a novel connection between these transport problems and issues of information asymmetry in finance. In the second part of the proposal, infinite-dimensionality arises from path dependence. The objective is to establish an Ishii's lemma applicable in this context and utilize this result to achieve well-posedness for two classes of second-order parabolic partial differential equations in the space of paths. The proposed research is expected to yield a comparison result for semicontinuous viscosity solutions of these PDEs, which directly impacts the convergence of numerical schemes for the hedging problem with rough volatility. In terms of applications, the infinite-dimensionality in this context stems from stochastic Volterra integral equations, time-inconsistent optimal control problems, and functional Itô calculus. 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.