Well-posedness and propagation of chaos for McKean-Vlasov equations with jumps and locally Lipschitz coefficients

TL;DR

This paper establishes the well-posedness and propagation of chaos for McKean-Vlasov equations with jumps and locally Lipschitz coefficients, extending classical results to more general coefficient conditions.

Contribution

It introduces a novel approach using truncation and Osgood's lemma to prove existence and uniqueness under locally Lipschitz conditions, overcoming technical challenges.

Findings

Proved strong well-posedness of McKean-Vlasov equations with locally Lipschitz coefficients.

Established propagation of chaos in this setting.

Developed a convergence proof for Picard iterations in distribution.

Abstract

We study McKean-Vlasov equations where the coefficients are locally Lipschitz continuous. We prove the strong well-posedness and a propagation of chaos property in this framework. These questions can be treated with classical arguments under the assumptions that the coefficients are globally Lipschitz continuous. In the locally Lipschitz case, we use truncation arguments and Osgood's lemma instead of Gr\"onwall's lemma. This approach entails technical difficulties in the proofs, in particular for the existence of solution of the McKean-Vlasov equations that are considered. This proof relies on a Picard iteration scheme that is not guaranteed to converge in an $L^1-$sense because the coefficients are not Lipschitz continuous. However, we still manage to prove its convergence in distribution, and the (strong) well-posedness of the equation using a generalization of Yamada and Watanabe results.