McKean-Vlasov Ito-Skorohod equations, and nonlinear diffusions with discrete jump sets

We consider a 'nonlinear' McKean-Vlasov Ito-Skorohod SDE, and develop a L1 contraction scheme so as to get good results on the non-compensated jumps. We prove existence and uniqueness results under natural Lipschitz assumptions. We show that a wide class of nonlinear martingale problems, giving most diffusions with discrete jump sets, can be represented by SDEs satisfying our L1 assumptions, but not more classical L2 ones. We use this on a probabilistic model for a chromatographic tube. We finish by a propagation of chaos result on sample-paths.