Forward–backward stochastic differential equations with nonsmooth coefficients

Solvability of forward–backward stochastic differential equations with nonsmooth coefficients is considered using the Four-Step Scheme and some approximation arguments. For the one-dimensional case, the existence of an adapted solution is established for the equation which allows the diffusion in the forward equation to be degenerate. As a byproduct, we obtain the existence of a viscosity solution to a one-dimensional nonsmooth degenerate quasilinear parabolic partial differential equation.