Propagation of Chaos for Derivatives of McKean-Vlasov Stochastic Differential Equations and Applications

TL;DR

This paper extends propagation of chaos results to derivatives of McKean-Vlasov SDEs, providing explicit convergence rates and analyzing derivatives with respect to initial conditions and noises.

Contribution

It proves convergence of derivatives of particle systems to those of McKean-Vlasov SDEs, with sharp explicit rates, enhancing understanding of sensitivity to initial data.

Findings

Convergence of derivatives with respect to initial perturbations.

Explicit sharp convergence rates established.

Intrinsic derivatives of particle systems converge to those of McKean-Vlasov SDEs.

Abstract

As an enhanced version of existing results on Kac's propagation of chaos, which describes the convergence of mean-field particle systems to a system of independent McKean-Vlasov particles as the number of particles tends to infinity, we prove the convergence at the level of derivatives with respect to the perturbations of the initial values and the driving noises, together with explicit convergence rates that can be sharp. As a consequence, the intrinsic derivative with respect to the initial distribution of a single particle converges to that of an independent McKean-Vlasov SDE.