Our study delves into the existence and regularity of  transmission problems represented by the equation $\sigma_i (|Du|) F(D^2u)=f$, concerning regions where $u > 0$ and $u < 0$. This work solves  the question on existence of solutions for the problem by utilizing a comparison principle for an auxiliary problem and Perron's Method.  Furthermore, we establish $C^1$ regularity estimates for the interior as a subsequent step in our analysis.