  In this paper we investigate a class of decoupled forward-backward SDEs,
where the volatility of the FSDE is degenerate and the terminal value of the
BSDE is a discontinuous function of the FSDE. Such an FBSDE is associated with
a degenerate parabolic PDE with discontinuous terminal condition. We first
establish a Feynman-Kac type representation formula for the spatial derivative
of the solution to the PDE. As a consequence, we show that there exists a
stopping time \tau such that the martingale integrand of the BSDE is continuous
before \tau and vanishes after \tau. However, it may blow up at \tau, as
illustrated by an example. Moreover, some estimates for the martingale
integrand before \tau are obtained. These results are potentially useful for
pricing and hedging discontinuous exotic options (e.g., digital options) when
the underlying asset's volatility is small, and they are also useful for
studying the rate of convergence of finite-difference approximations for
degenerate parabolic PDEs.
