Coupled FBSDEs with Measurable Coefficients and its Application to Parabolic PDEs

Abstract

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) $$\begin{array}{rl}& \mathcal{L}u(t,x)+F(t,x,u,{\partial }_{x}u)=0;u(T,x)=h(x)\\ & \mathcal{L}:={\partial }_t+\frac{1}{2}\sum _{i,j=1}^m(\sigma {\sigma }^{⊺}{)}_{ij}(t,x){\partial }_{x_i{x}_j}^2\end{array}$$ 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.

24 page