\] Assume that $${\left\|\xi\right\|}_\infty\leq\frac{0}{0\sqrt{00}\sqrt{0000}\mu e^{\frac00CT}},$$ where $C$ is the Lipschitz constant of $g$ in $y$, and $\mu$ is the constant appearing in Assumption \[lipschitz\_assumption\]. Then under Assumption \[lipschitz\_assumption\] with $g(0,0,0)=0$, there exists a unique solution $(Y,Z,U)\in\mathcal S^\infty\times\mathbb H^0_{\rm{BMO}}\times\mathbb J^0_{\rm{BMO}}\cap L^\infty(\nu)$ of the BSDEJ .  