An algebraic convergence rate for the optimal control of McKean-Vlasov dynamics

TL;DR

This paper proves an algebraic convergence rate for the value functions of N-particle stochastic control problems to the mean field control problem, even with common noises and non-smooth value functions.

Contribution

It introduces a novel algebraic convergence rate for large N limits in mean field control with both idiosyncratic and common noises.

Findings

Established algebraic convergence rate in N-particle control problems

Extended convergence analysis to non-smooth value functions

Utilized Lipschitz, semi-concavity estimates, and concentration inequalities

Abstract

We establish an algebraic rate of convergence in the large number of players limit of the value functions of N-particle stochastic control problems towards the value function of the corresponding McKean-Vlasov problem also known as mean field control. The rate is obtained in the presence of both idiosyncratic and common noises and in a setting where the value function for the McKean-Vlasov problem need not be smooth. Our approach relies crucially on uniform in N Lipschitz and semi-concavity estimates for the N-particle value functions as well as a certain concentration inequality.