  Using purely probabilistic methods, we prove the existence and the uniqueness
of solutions fora system of coupled forward-backward stochastic differential
equations (FBSDEs) with measurable, possibly discontinuous coefficients. As a
corollary, we obtain the well-posedness of semilinear parabolic partial
differential equations (PDEs) $$ \begin{aligned} &\mathcal{L}
u(t,x)+F(t,x,u,\partial_x u)=0;\qquad u(T,x)=h(x)\\
&\mathcal{L}:=\partial_t+\frac{1}{2}\sum_{i,j=1}^m(\sigma\sigma^\intercal)_{ij}(t,x)\partial^2_{x_ix_j}
\end{aligned} $$ in the natural domain of the second-order linear parabolic
operator $\mathcal{L}$. We allow $F$ and $h$ to be discontinuous with respect
to $x$. Finally, we apply the result to optimal policy-making for pandemics and
pricing of carbon emission financial derivatives.
