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 a consequence we can then argue that the FBSDE admits a weak solution in the sense of [1 Antonelli, F., Ma, J. (2003). Weak solutions of forward-backward SDE’s. Stochastic Analysis and Applications 21(3):493–514.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar], 2 Ma, J., Zhang, J., Zheng, Z. (2008). Weak solutions for backward stochastic differential equations, A martingale approach. The Annals of Probability 36(6):2092–2125.[Crossref], [Web of Science ®] , [Google Scholar]]. 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.