A note on L\'evy-driven McKean-Vlasov SDEs under monotonicity

TL;DR

This paper establishes the strong well-posedness and propagation of chaos for McKean-Vlasov SDEs driven by Lévy processes under weak monotonicity and coercivity conditions, filling a long-standing gap in the literature.

Contribution

It provides the first comprehensive analysis of strong well-posedness for Lévy-driven McKean-Vlasov SDEs under monotonicity, using interlacing and fixed point techniques.

Findings

Proved strong well-posedness of Lévy-driven McKean-Vlasov SDEs under weak monotonicity.

Established weak and strong propagation of chaos for these equations.

Extended results to cases with common noise and conditional propagation of chaos.

Abstract

In this note, under a weak monotonicity and a weak coercivity, we address strong well-posedness of McKean-Vlasov stochastic differential equations (SDEs) driven by L\'{e}vy jump processes, where the coefficients are Lipschitz continuous (with respect to the measure variable) under the $L^\beta$-Wasserstein distance for $\beta\in[1,2].$ Moreover, the issue on the weak propagation of chaos (i.e., convergence in distribution via the convergence of the empirical measure) and the strong propagation of chaos (i.e., at the level paths by coupling) is explored simultaneously. To treat the strong well-posedness of McKean-Vlasov SDEs we are interested in, we investigate strong well-posedness of classical time-inhomogeneous SDEs with jumps under a local weak monotonicity and a global weak coercivity. Such a result is of independent interest, and, most importantly, can provide an available reference on strong well-posedness of L\'{e}vy-driven SDEs under the monotone condition, which nevertheless is missing for a long time. Based on the theory derived, along with the interlacing technique and the Banach fixed point theorem, the strong well-posedness of McKean-Vlasov SDEs driven by L\'{e}vy jump processes can be established. Additionally, as a potential extension, strong well-posedness and conditional propagation of chaos are treated for L\'{e}vy-driven McKean-Vlasov SDEs with common noise under a weak monotonicity.