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 .
In this article, we prove the existence of bounded solutions of quadratic backward SDEs with jumps, that is to say for which the generator has quadratic growth in the variables (z,u). From a technical point of view, we use a direct fixed point approach as in Tevzadze [ 38 ] , which allows us to obtain existence and uniqueness of a solution when the terminal condition is small enough. Then, 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 .
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which allows us to obtain existence and uniqueness of a solution when the terminal condition is small enough. Then,
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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
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