Mathematics – Probability
Scientific paper
2005-08-24
Annals of Applied Probability 2005, Vol. 15, No. 3, 1798-1831
Mathematics
Probability
Published at http://dx.doi.org/10.1214/105051605000000232 in the Annals of Applied Probability (http://www.imstat.org/aap/) by
Scientific paper
10.1214/105051605000000232
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.
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