Stochastic Portfolios, Controls, Interacting Particles and Hereditary Limits

Columbia University, New York NY

Investigators

Abstract

This project investigates foundational and emerging areas of Probability Theory, with broad applications in mathematics, physics, finance, and machine learning. A central goal is to advance theoretical understanding while supporting the training and development of future researchers in this dynamic field. Key areas of focus include the foundations of limit theorems, entropic gradient flows, stochastic control with partial observation, and stochastic portfolio theory. The research integrates tools from stochastic analysis, functional analysis, and information theory, combining mathematical rigor with practical relevance. It aims to advance the theoretical foundations of Probability while offering insights relevant to real-world systems. The project aims to develop a stochastic theory of portfolios that operates without equivalent martingale measures and allows for portfolio outperformance. Grounded in local martingale deflators, the optional decomposition theorem, and functional analysis, this framework excludes only extreme forms of arbitrage and provides a rigorous foundation for hedging and portfolio optimization. It will address diverse market models, including those with infinitely many assets, open markets with dynamic composition, portfolio constraints, and model uncertainty. The project also advances stochastic optimization under partial information, combining control, stopping, and filtering -- areas with limited existing theory. Building on recent progress by the principal investigator’s group, it introduces a new trajectorial/perturbation approach to Otto calculus via time-reversal and optimal transport, with applications to McKean-Vlasov diffusions, neural network training, and entropy flows in Markov processes. Finally, the project will characterize necessary and sufficient conditions for lacunary and hereditary versions of classical probabilistic limit theorems, using novel approximation techniques such as exchangeable subsequences and Egorov-type methods. 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.