Representation of solutions to BSDEs associated with a degenerate FSDE

Abstract

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 τ such that the martingale integrand of the BSDE is continuous before τ and vanishes after τ. However, it may blow up at τ, as illustrated by an example. Moreover, some estimates for the martingale integrand before τ 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.