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

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.

Regularity of the value function and quantitative propagation of chaos for mean field control problems

“…We finally comment about possible extensions to problems with common noise emphasizing once more that we do not assume any monotonicity/convexity. The convergence (with algebraic rate) of V N to U for MFC problems with common noise was established in [6]. However, the generalization of the results of the present paper to such a setting is far from clear.…”

“…The estimates of m and the local regularity of u are standard. Indeed, the uniform bound on Du follows as in the proof of Lemma 3.3 in [6], and then the estimate on m are immediate. Moreover, the local regularity of u is a consequence of the classical parabolic regularity theory.…”

“…This is in contrast with the limit U, which might not be C 1 . The following result, proved in [6], states however that both maps are uniformly Lipschitz continuous.…”

“…Several other results on the convergence of MFC problems without diffusion were obtained in Cavagnari, Lisini, Orrieri and Savaré [8] and Gangbo, Mayorga and Swiech [14]. A quantitative convergence rate for the value function V N to U was given, for problems on a finite state space, in Kolokoltsov [17] and Cecchin [9] and, for problems on the continuous state space, in Baryaktar and Chakraborty [1] under a certain structural dependence of the data on the measure variable, in Germain, Pham and Warin [15] under the assumption that the limit value is smooth, and in Cardaliaguet, Daudin, Jackson and Souganidis [6] under a decoupling assumption on the Hamiltonian. In addition, a propagation of chaos is proved in [1,9,15] assuming, however, that the limit value function is smooth.…”

“…there exists a constant C > 0 such that, for all (x, p The assumptions on the Hamiltonian H are fairly standard, although a little restrictive, and are used in [6] to obtain, independent of N , Lipschitz estimates on the value function V N ; see Lemma 1.7. An example satisfying (1.5) is a Hamiltonian of the form H(x, p) = |p| 2 + V (x) • p for some smooth and globally Lipschitz continuous vector field V : R d → R d .…”