UCSB Student Seminar in Financial Mathematics

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).