  We establish well-posedness for a class of systems of SDEs with non-Lipschitz
coefficients in the diffusion and jump terms and with two sources of
interdependence: a monotone function of all the components in the drift of each
SDE and the correlation between the driving Brownian motions and jump random
measures. Pathwise uniqueness is derived by employing some standard techniques.
Then, we use a comparison theorem along with our uniqueness result to construct
non-negative, $L^1$-integrable c\`adl\`ag solutions as monotone limits of
solutions to approximating SDEs, allowing for time-inhomogeneous drift terms to
be included. Our approach allows also for a comparison property to be
established for the solutions to the systems we investigate. The applicability
of certain systems in financial modeling is also discussed.
