In this paper, we study a class of quadratic-exponential growth BSDEs with jumps. The quadratic structure was introduced by Barrieu & El Karoui (2013) and yields a very useful universal bound on the possible solutions. With the bounded terminal condition as well as an additional local Lipschitz continuity, we give a simple and streamlined proof for the existence and the uniqueness of the solution . The universal bound and the stability result for the locally Lipschitz BSDEs with coefficients in the BMO space enable us to show the strong convergence of a sequence of globally Lipschitz BSDEs . The result is then used to generalize the existing results on the Malliavin's differentiability of the quadratic BSDEs in the diffusion setup to the quadratic-exponential growth BSDEs with jumps .
We investigate a class of quadratic-exponential growth BSDEs with jumps. The quadratic structure introduced by Barrieu & El Karoui (2013) yields the universal bound on the possible solutions. With a bounded terminal condition and local Lipschitz continuity, we give a simple and streamlined proof for the existence as well as the uniqueness of the solution without using the comparison principle. The properties of locally Lipschitz BSDEs with coefficients in BMO space enable us to show the strong convergence of a sequence of globally Lipschitz BSDEs to the interested one, which is then used to give sufficient conditions for the Malliavin's differentiability .
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