Well-posedness and propagation of chaos for L{\'e}vy-driven McKean-Vlasov SDEs under Lipschitz assumptions

TL;DR

This paper establishes the well-posedness and propagation of chaos for Lévy-driven McKean-Vlasov SDEs under Lipschitz conditions, and improves convergence rates for certain stable processes.

Contribution

It proves strong well-posedness and enhances convergence rates for Lévy-driven McKean-Vlasov SDEs with linear interactions and stable noise.

Findings

Proved strong well-posedness under Lipschitz assumptions.

Established quantitative propagation of chaos for the particle system.

Improved convergence rates for linear interactions with stable noise.

Abstract

The first goal of this note is to prove the strong well-posedness of McKean-Vlasov SDEs driven by L{\'e}vy processes on $\mathbb{R}^d$ having a finite moment of order $\beta \in [1,2]$ 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 $\alpha$-stable process with $\alpha \in (1,2)$, for which we have $\beta < \alpha$.