Existence of solutions to a system of SDEs with mean-field drift and jump random measures

We consider the well-posedness of a system of multi-dimensional SDEs which are correlated through a non-homogeneous mean-field term in each drift and also by driving Brownian motions and jump random measures. Supposing the drift coefficients are non-Lipschitz, we prove for the system the existence of strong, $L^1$-integrable, cadlag solutions which can be obtained as monotone limits of solutions to some approximating systems, extending existing results for one-dimensional jump SDEs with non-Lipschitz coefficients. We show in addition that the solutions are positive.