Coupled FBSDEs with measurable coefficients and its application to parabolic PDEs

Using purely probabilistic methods, we prove the existence and uniqueness of solutions for a system coupled forward-backward stochastic differential equations (FBSDEs) with measurable, possibly discontinuous, coefficients. As corollary, obtain well-posedness semilinear parabolic partial (PDEs)Lu(t,x)+F(t,x,u,∂xu)=0;u(T,x)=h(x) in natural domain second order linear operator L when F h are not necessarily continuous respect to x. We also provide sufficient condition this solution be Sobolev space. Finally, apply result optimal policymaking pandemics pricing carbon emission allowances.