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.