Before revision:  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 .
After revision:  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 to the interested one, which 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 .