Before revision: We prove the existence of bounded solutions of quadratic backward SDEs with jumps, using a direct fixed point approach as in Tevzadze [ 36 ] . Under an additional standard assumption, we prove a uniqueness result ,  thanks to a comparison theorem. Then we study the properties of the corresponding g-expectations, we obtain in particular a non linear Doob-Meyer decomposition for g-submartingales and their regularity in time. As a consequence of this results, we obtain a converse comparison theoremfor our class of BSDEs. We give applications for dynamic risk measures and their dual representation, and compute their inf-convolution, with some explicit examples .
After revision: We prove the existence of bounded solutions of quadratic backward SDEs with jumps, using a direct fixed point approach as in Tevzadze [ 36 ] . Under an additional standard assumption, we prove a uniqueness result , thanks to a well-chosen splitting, we recover an existence result for general bounded solution. Under additional assumptions, we can obtain stability results and a comparison theorem, which as usual implies uniqueness .