Before revision:  In particular, we obtain existence, comparison, and stability results . Moreover, we study the link between the reflected forward backward SDEs and the obstacle problem for semi-linear parabolic PDEs in which the non-linearity appears as the square of the gradient.
After revision:  In particular, we obtain existence, comparison, and stability results , and consider the optimal stopping for quadratic g-evaluations. As an application of our results we analyze the obstacle problem for semi-linear parabolic PDEs in which the non-linearity appears as the square of the gradient.