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

We study McKean–Vlasov equations where the coefficients are locally Lipschitz continuous. We prove the strong well-posedness and a propagation of chaos property. These questions are classical under the assumptions that the coefficients are Lipschitz continuous. In the locally Lipschitz case, we use truncation arguments and Osgood's lemma instead of Grönwall's lemma. Technical difficulties appear in the proofs, in particular for the existence of solution of the McKean–Vlasov equations. This proof relies on a Picard iteration scheme that is not guaranteed to converge in an L¹−sense. However, we prove its convergence in distribution, and the (strong) well-posedness of the equation.