Market Microstructure and Regularized Optimal Transport

This project develops applied mathematics in two areas: comparing complex data sets and understanding how modern financial markets operate. Optimal transport is the mathematics of moving goods, images, or data from one configuration to another at the lowest cost. It is used in machine learning, image processing, economics, and related areas. The computational modifications that make optimal transport usable at scale can blur the result or become unstable when high accuracy is required. This project studies leading regularization methods and determines when they are accurate and reliable. The project also studies how large trades reveal private information through price movements and how decentralized cryptocurrency exchanges compete through trading fees. Better models can reduce execution costs for institutional investors such as pension funds, improve the efficiency and resilience of financial infrastructure, and clarify risks in digital-asset markets. Methods for limiting information leakage may also apply to strategic planning and technology deployment. By advancing fundamental mathematics, developing algorithms, and connecting theory to real market design, the project promotes technological leadership and economic prosperity. The project will also train postdoctoral, graduate, and undergraduate researchers and strengthen the STEM workforce through research, courses, and interdisciplinary meetings. The first research thrust investigates entropically regularized optimal transport for Monge’s distance cost, which defines the Kantorovich–Wasserstein distance. As the regularization parameter tends to zero, the unregularized problem can have many optimal transport plans. The project aims to derive a second-order asymptotic expansion of the regularized cost and resolve the entropic selection conjecture by identifying the specific plan selected in the limit. The approach combines disintegration along transport rays, new regularity estimates for Kantorovich potentials, and quantitative constructions of approximate couplings. The second thrust studies quadratically regularized optimal transport for squared-distance cost. Its central goal is to establish uniform Hessian bounds for the dual potentials. Geometric analysis of support sections will connect strong convexity, C1 convergence, and the sharp rate at which regularized plans concentrate around Brenier’s map. The third thrust develops partial-information execution games in which an arbitrageur infers an executor’s private order size from observed prices. Using stochastic filtering and variational analysis, the project will characterize optimal control of information leakage, the role of volatility, and the resulting order-flow and price dynamics. The fourth thrust studies dynamic competition among multiple automated market makers and centralized exchanges. Large-scale simulation and complementary continuous-time analysis will be used to analyze equilibrium fee schedules, order flows, and liquidity provision. Additional projects will develop risk-neutral pricing and arbitrage bounds for perpetual futures with nonlinear funding rules. 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.