  In a previous work, P. Briand and Y. Hu proved the uniqueness among the
solutions which admit every exponential moments. In this paper, we prove that
uniqueness holds among solutions which admit some given exponential moments.
These exponential moments are natural as they are given by the existence
theorem. Thanks to this uniqueness result we can strengthen the nonlinear
Feynman-Kac formula proved by P. Briand and Y. Hu.
