Rate of convergence for particle approximation of PDEs in Wasserstein space

Abstract

We prove a rate of convergence of order 1/N for the N -particle approximation of a second-order partial differential equation in the space of probability measures, like the Master equation or Bellman equation of mean-field control problem under common noise. The proof relies on backward stochastic differential equations techniques.

Importance Sampling for the Empirical Measure of Weakly Interacting Diffusions

“…The reason for the assumptions in Theorem 2.3 being stated as such is that little is available in the current literature in terms of sufficient conditions for classical solutions to Equation ( 15), let alone on obtaining a the desired formal expansions in N . Indeed, even the convergence of Φ N to φ 0 (corresponding to the first order expansion) has only been studied in the case where the particles have common noise -see [54].…”

“…Rigorous justifications for this limit in the case where there is a common driving noise between the particles (1) can be found in [54].…”

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

“…See also [17] for more on what convergence results can be obtained once (1.8) has a sufficiently smooth solution. This argument is similar to the approach taken in [5,7] to study the convergence problem in the context of mean field games (see Lasry and Lions [24]) in situations where a classical solution to the so-called master equation is known to exist; also see Bayraktar and Cohen [1] and Cecchin and Pelino [10] for related results.…”

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

“…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.…”