Past events

Abstract:
Rough path theory provides a natural pathwise framework for continuous non-semimartingale models, making it attractive for financial models with memory or rough signals. Yet in no-arbitrage theory, trading gains and self-financing conditions depend on the chosen notion of integration. In this talk, we study whether rough-path-based market models can support a meaningful no-free-lunch theory beyond the classical Itô/semimartingale setting.

We introduce a No Controlled Free Lunch (NCFL) condition and reduce it to the unbiasedness of the price driver as a rough integrator under an equivalent measure. Our main contribution is a classification of such unbiased rough integrators for progressively richer classes of controlled portfolios: as admissible strategies expand from Markovian to signature-type portfolios, viable rough-path market models are forced first toward Gaussian-Hermite rough paths and ultimately toward Brownian Itô lifts, up to a time change. Thus, frictionless rough-path market models with sufficiently rich strategy classes are pushed back toward the classical semimartingale paradigm.

The talk will consist of three parts: a gentle introduction to rough path theory, a review of classical no-arbitrage and the Kreps–Yan theorem, and the rough-path counterpart together with its financial implications.

This is joint work with Tomoyuki Ichiba. 

Abstract:
Mean-field control and games with common noise provide a powerful framework for modeling the collective behavior of large populations under shared randomness, such as systemic risk in finance and aggregate environmental shocks in economics. One of the central problems in mean-field problems is to solve the associated McKean–Vlasov forward-backward stochastic differential equations (MV-FBSDEs). However, most existing methods are tailored to special settings in which the mean-field interaction depends only on expectations or other low-order moments, and are therefore inadequate for problems with fully distribution-dependent interactions.

In our presentation, we propose a deep learning-based algorithm for solving MV-FBSDEs with general mean-field interactions, both with and without common noise. Building on the idea of fictitious play, our method iteratively solves conditional FBSDEs for fixed distributions, while updating the distributional dependence through supervised learning. In particular, when it comes to MV-FBSDEs with common noise, we use path signatures to approximate the conditional distribution along the common noise trajectory. Deep neural networks are employed both to solve the resulting FBSDEs and to approximate the distribution-dependent coefficients, making the method scalable to high-dimensional problems. Under suitable assumptions, we establish convergence of the fictitious play iteration, with the overall error controlled by the supervised learning approximation error.

This is joint-work with Ruimeng Hu (UCSB), Mathieu Laurière (NYU Shanghai) and Jiacheng Zhang (CUHK).

Abstract:

Probabilistic representations provide a particle viewpoint to studying evolution equations.  For continuous-time dynamics, we review how Dynkin’s formula and Itô calculus lead to probabilistic representations of solutions to PDEs in terms of Markov processes with both continuous and jump-type potentials. In particular we derive the  probabilistic representation of the one-dimensional linear Dirac equation and show that it arises as a scaling limit of the discrete-time quantum walk via a Feynman-type formula, derived using a Poissonization trick.

Abstract:

Mean-field games (MFGs) provide elegant approximations to stochastic differential games with a large number of infinitesimal players and have become increasingly important in financial applications. In this talk, we introduce the mean-field actor-critic flow (MFAC), a reinforcement learning (RL)-based algorithm designed to solve MFGs, and establish its validity through both theoretical analysis and numerical experiments.

Unlike traditional RL approaches, our method employs partial differential equations (PDEs) to model the continuous-time training dynamics of the actor, critic, and population distribution. Inspired by optimal transport, the distributional dynamics evolve along a novel geodesic flow, and we prove convergence under a single time scale. Furthermore, borrowing ideas from generative modeling, we parameterize the distribution via score functions, the sampling from which is supported by Langevin Monte Carlo. Through numerical experiments, we demonstrate that MFAC effectively solves MFGs, achieving state-of-the-art performance.

This is joint work with Ruimeng Hu (UCSB) and Mo Zhou (UCLA).

Abstract:

Concentration of measure refers to the phenomenon that functions of independent random variables are sharply concentrated around their expected values. In this talk, we introduce the Hanson–Wright inequality, a class of concentration inequalities for quadratic forms of sub-Gaussian random vectors. We begin by reviewing classical concentration results, including Gaussian concentration and Bernstein’s inequality, derived via the Chernoff bound. We then present the Hanson–Wright inequality and outline its proof using Gaussian replacement and decoupling techniques. Finally, we discuss several applications of this inequality in machine learning and high-dimensional data analysis.