Well-posedness and propagation of chaos for L{é}vy-driven McKean-Vlasov SDEs under Lipschitz assumptions
Thomas Cavallazzi
TL;DR
The paper establishes strong well-posedness for Lévy-driven McKean–Vlasov SDEs with finite $β$-moments under Lipschitz conditions, using a Banach fixed-point approach on the path-space of probability measures and a separation of small and large jumps. It then analyzes the associated mean-field particle system to obtain quantitative strong propagation of chaos, providing explicit rates that depend on dimension, moment index, and the noise type; notably, rates improve for linear interactions and for $α$-stable noise with $α∈(1,2)$, where $β<α$ is required. The methods combine fixed-point arguments, careful handling of jump components, and moment/Mass transportation estimates, yielding a comprehensive picture of how finite-N systems approximate the McKean–Vlasov limit in Lévy-driven settings. These results extend the Brownian framework to more general jump-driven dynamics and offer precise convergence rates that inform both theoretical analysis and numerical simulations of interacting particle systems.
Abstract
The first goal of this note is to prove the strong well-posedness of McKean-Vlasov SDEs driven by L{é}vy processes on $\mathbb{R}^d$ having a finite moment of order $β\in [1,2]$ and under standard Lipschitz assumptions on the coefficients. Then, we prove a quantitative propagation of chaos result at the level of paths for the associated interacting particle system, with constant diffusion coefficient. Finally, we improve the rates of convergence obtained for linear interactions with respect to the measure and when the noise is a $α$-stable process with $α\in (1,2)$, for which we have $β< α$.