Extended mean field control problem: a propagation of chaos result

Abstract

In this paper, we study the extended mean field control problem, which is a class of McKean-Vlasov stochastic control problem where the state dynamics and the reward functions depend upon the joint (conditional) distribution of the controlled state and the control process. By considering an appropriate controlled Fokker-Planck equation, we can formulate an optimization problem over a space of measure-valued processes and, under suitable assumptions, prove the equivalence between this optimization problem and the extended mean-field control problem. Moreover, with the help of this new optimization problem, we establish the associated limit theory i.e. the extended mean field control problem is the limit of a large population control problem where the interactions are achieved via the empirical distribution of state and control processes.

Mean Field Games of Controls: on the convergence of Nash equilibria

“…and in some way by Lacker [22]. Borrowing techniques from [21], under suitable assumptions, we prove that the sequence of empirical measure flows (ϕ N,X , ϕ N ) is tight in a suitable space, and with the help of techniques introduced in our companion paper [10], we show that every limit in distribution is a measure-valued mean field equilibrium. And conversely, for each measure-valued mean field equilibrium, we construct an approximate Nash equilibrium which has this measure-valued mean field equilibrium as limit.…”

“…Remark 2.2. Most of these assumptions are classical in the study of mean field games and control problems (see Lacker [21], Djete, Possamaï, and Tan [12] and Djete [10] ). Only the "separability condition" and the "non-degeneracy condition" can be seen as non-standard.…”

“…3)) and the discussion in [10], we carefully formulate the notion of measure-valued control rules which is essential for the notion of measure-valued MFG equilibrium that will be introduced just after.…”

“…Remark 2.9. Looking at this kind of measure-valued solution is largely inspired by the notion considered in [10] in the McKean-Vlasov setting. However, our notion of (ǫ-) measure-valued MFG solution enters completely in the framework of MFG solutions considered in Carmona, Delarue, and Lacker [8].…”

“…Despite many differences, this article is in the same spirit as [21], which is, in the framework without law of control, the most significant paper investigating the connection between large population differential games and the MFG under very general assumptions. We want to emphasize that the interesting techniques developed in [21] do not work in the case of MFG of controls, in the presence of the law of control, the assumptions of continuity on the coefficients are no longer verified (see also discussion in Djete [10]).…”