Abstract: In this talk, I will discuss a forthcoming joint work with Alekos Cecchin, Samuel Daudin, and Mattia Martini, in which we obtain quantitative convergence results for mean field control in the presence of common noise and degenerate idiosyncratic noise. In particular, we estimate the speed with which the value functions $V^N$ for certain finite-dimensional control problems converge towards the value function $U$ of a corresponding mean field control problem. The main difficulty in this setting is that the limiting value function $U$ is not smooth, and is at best Lipschitz with respect to the 1-Wasserstein distance. Our proof strategy is to approximate $U$ by functions which are smoother, and are almost subsolutions to the appropriate infinite dimensional Hamilton-Jacobi equation. In the case that the noises are constant (but possibly degenerate), we obtain the nearly optimal rate $N^{-1/(d+7)}$, while in the case of degenerate and non-constant volatility (which turns out to be much trickier) we obtain the rate $N^{-1/(3d + 19)}$. 

Brief bio: Joe is a Dickson Instructor and NSF Postdoctoral Research Fellow at the University of Chicago. He received his PhD in 2023 from the University of Texas at Austin. His research interests are in partial differential equations, stochastic analysis, and their applications to stochastic control and stochastic differential games.