Forward-backward SDEs with Discontinuous Coefficients

In this paper we are interested in the well-posedness of a class of fully coupled forward-backward SDE (FBSDE) in which the forward drift coefficient is allowed to be discontinuous with respect to the backward component of the solution. Such an FBSDE is motivated by a practical issue in regime-switching term structure interest rate models, and the discontinuity makes it beyond any existing framework of FBSDEs. In a Markovian setting with non-degenerate forward diffusion, we show that a decoupling function can still be constructed and that it is a Sobolev solution to the corresponding quasilinear PDE. As as a consequence we can then argue that the FBSDE admits a weak solution in the sense of [2] and [29]. In the one-dimensional case, we further prove that the weak solution of the FBSDE is actually strong, and it is pathwisely unique. Our approach does not use the well-known Yamada-Watanabe Theorem, but instead follows the idea of Krylov for SDEs with measurable coefficients.