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.
Categories: math.PR