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 for a 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)Lu(t,x)+F(t,x,u,∂xu)=0;u(T,x)=h(x) in the natural domain of the second order linear parabolic operator L when F and h are not necessarily continuous with respect to x. We also provide a sufficient condition for this solution to be in a Sobolev space. Finally, we apply the result to optimal policymaking for pandemics and pricing of carbon emission allowances.