A stochastic Gronwall lemma and well-posedness of path-dependent SDEs driven by martingale noise

We show existence and uniqueness of solutions stochastic path-dependent differential equations driven by cadlag martingale noise under joint local monotonicity coercivity assumptions on the coefficients with a bound in terms supremum norm. In this set-up, usual proof using ordinary Gronwall lemma together Burkholder-Davis-Gundy inequality seems impossible. order to solve problem, we prove new quite general for martingales Lenglart's inequality.