Jumping SDEs: absolute continuity using monotonicity

We study the solution X={Xt}t∈[0,T] to a Poisson-driven SDE. This equation is “irregular” in the sense that one of its coefficients contains an indicator function, which allows to generalize the usual situations: the rate of jump of X may depend on X itself. For t>0 fixed, the random variable Xt does not seem to be differentiable (with respect to the alea) in a usual sense (see e.g. Séminaire de Probabilités XVII, Lecture Notes in Mathematics, Vol. 986, Springer, Berlin, 1983, pp. 132–157), and actually not even continuous. We thus introduce a new technique, based on a sort of monotony of the map ω↦Xt(ω), to prove that under quite stringent assumptions, which make possible comparison theorems, the law of Xt admits a density with respect to the Lebesgue measure on R.